influence of indentations on rolling bearing...

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n d’ordre : 2008-ISAL-0054 Ann´ ee 2008 THESE pr´ esent´ ee DEVANT L’INSTITUT NATIONAL DES SCIENCES APPLIQUEES DE LYON pour obtenir LE GRADE DE DOCTEUR ECOLE DOCTORALE DES SCIENCES POUR L’INGENIEUR DE LYON : MECANIQUE, ENERGETIQUE, GENIE CIVIL, ACOUSTIQUE (M.E.G.A.) SPECIALITE : MECANIQUE par Nans BIBOULET Ing´ enieur I.N.S.A. de Lyon INFLUENCE OF INDENTATIONS ON ROLLING BEARING LIFE Soutenue le 16 septembre 2008 devant la commission d’examen : Jury : M. K. Dang Van M. J. Frˆ ene Rapporteur M. C.J. Hooke Rapporteur M. L. Houpert M. A.A. Lubrecht M. J. Seabra M. F. Ville Cette th` ese a ´ et´ e pr´ epar´ ee au Laboratoire de M´ ecanique des Contacts et des Structures de l’I.N.S.A. de Lyon

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Page 1: Influence of indentations on rolling bearing lifetheses.insa-lyon.fr/publication/2008ISAL0054/these.pdf · INFLUENCE OF INDENTATIONS ON ROLLING BEARING LIFE Soutenue le 16 septembre

n d’ordre : 2008-ISAL-0054 Annee 2008

THESE

presentee

DEVANT L’INSTITUT NATIONAL DES SCIENCES APPLIQUEES

DE LYON

pour obtenir

LE GRADE DE DOCTEUR

ECOLE DOCTORALE DES SCIENCES POUR L’INGENIEUR DE LYON :MECANIQUE, ENERGETIQUE, GENIE CIVIL, ACOUSTIQUE (M.E.G.A.)

SPECIALITE : MECANIQUE

par

Nans BIBOULET

Ingenieur I.N.S.A. de Lyon

INFLUENCE OF INDENTATIONS ON ROLLING

BEARING LIFE

Soutenue le 16 septembre 2008 devant la commission d’examen :

Jury : M. K. Dang VanM. J. Frene RapporteurM. C.J. Hooke RapporteurM. L. HoupertM. A.A. LubrechtM. J. SeabraM. F. Ville

Cette these a ete preparee au Laboratoire de Mecanique des Contacts et des Structures

de l’I.N.S.A. de Lyon

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I

Resume

Les lubrifiants contiennent des particules qui sont a la fois initialementpresentes dans le lubrifiant et dans le mecanisme (fabrication, stockage. . . );generees durant le rodage, par usure ou corrosion; ou bien issues de sourcesexternes.

Les particules deteriorent les surfaces en creant des indents lorsqu’ellessont piegees dans les contacts. Ces indents augmentent le risque de rupturepar fatigue en induisant des perturbations de pression et de contraintes.

Ce travail est base sur une etude des contacts indentes secs et lubrifies(EHD). L’equation de Reynolds en fluide Newtonien est utilisee. Les tech-niques mutigrilles et multi-integration sont employees.

L’objectif de ce travail est de proposer un modele de prediction des per-turbations des champs de pression et de contraintes, et finalement un modelede reduction de duree de vie des contacts indentes en fonction de la geometriedes indents et des conditions de contact.

Mot-cles : contact ponctuel, contact elliptique, contact sec, lubrifica-tion, EHD, elastohydrodynamique, Reynolds, multigrilles, multi-integration,reduction d’amplitude, pollution, contamination, debris, particule, indent,indentation, prediction de duree de vie, roulement, fatigue

Abstract

All lubricants contain particles. These particles are both initially presentin the lubricant and mechanisms (manufacturing process, storage. . . ); gen-erated during running-in, through wear and corrosion; or come from externalsources.

These particles damage the surfaces, creating indents when they aresquashed in the contact. These indents increase the fatigue failure riskinducing pressure and stress perturbations.

This work is based on a dry and lubricated (EHL) contact study. TheReynolds equation for Newtonian fluids is used. Multigrid and multi-integra-tion techniques are used to limit computing time.

The aim is to predict pressure and stress distribution perturbations, andfinally to predict the life reduction of indented contacts as a function of theindent geometry and the operating conditions.

Keywords: point contact, elliptical contact, dry contact, lubrication,EHL, elastohydrodynamic, Reynolds, multigrid, multi-integration, ampli-tude reduction theory, pollution, contamination, debris, particle, indent,indentation, life prediction, rolling element bearing, fatigue

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II

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Foreword

III

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IV

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Contents

Notation 2

1 General introduction 5

1.1 Pollution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2 Modeling the pollution impact . . . . . . . . . . . . . . . . . 10

1.2.1 Lubricant pollution and surface indentation . . . . . . 10

1.2.2 Life prediction . . . . . . . . . . . . . . . . . . . . . . 13

1.3 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2 Modeling and numerical methods 17

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2 Dry contact . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.2.1 Gap height equation . . . . . . . . . . . . . . . . . . . 18

2.2.2 Force balance equation . . . . . . . . . . . . . . . . . . 20

2.3 EHL contact . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.3.1 Lubricant . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.3.2 Reynolds equation . . . . . . . . . . . . . . . . . . . . 23

2.4 Subsurface stress distribution . . . . . . . . . . . . . . . . . . 23

2.5 Indent geometry . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.5.1 Natural indentation observations . . . . . . . . . . . . 25

2.5.2 Indent creation modeling . . . . . . . . . . . . . . . . 27

2.5.3 Analytical description . . . . . . . . . . . . . . . . . . 27

2.6 Numerical techniques . . . . . . . . . . . . . . . . . . . . . . . 28

2.6.1 MLMI principles . . . . . . . . . . . . . . . . . . . . . 29

2.6.2 Multigrid principles . . . . . . . . . . . . . . . . . . . 30

2.6.3 Transient calculation implementation . . . . . . . . . . 35

2.6.4 Numerical accuracy . . . . . . . . . . . . . . . . . . . 36

2.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3 Dry and EHL pressure in indented contacts 45

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.2 Dry contact pressure . . . . . . . . . . . . . . . . . . . . . . . 46

3.2.1 Indent crossing . . . . . . . . . . . . . . . . . . . . . . 47

V

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VI CONTENTS

3.2.2 Ellipticity . . . . . . . . . . . . . . . . . . . . . . . . 483.2.3 Analytical model for dry contacts . . . . . . . . . . . 51

3.3 EHL contact pressure . . . . . . . . . . . . . . . . . . . . . . 583.3.1 Indent crossing . . . . . . . . . . . . . . . . . . . . . . 593.3.2 Ellipticity . . . . . . . . . . . . . . . . . . . . . . . . 603.3.3 Pure-rolling EHL pressure prediction . . . . . . . . . 623.3.4 Sliding influence . . . . . . . . . . . . . . . . . . . . . 72

3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4 Indented contact stress and life 774.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 774.2 Contact stress and life . . . . . . . . . . . . . . . . . . . . . . 78

4.2.1 Stress and life criteria . . . . . . . . . . . . . . . . . . 784.2.2 Pseudo-transient calculation . . . . . . . . . . . . . . 814.2.3 Stress and life prediction . . . . . . . . . . . . . . . . . 83

4.3 Experimental indentation and life calculation . . . . . . . . . 874.3.1 Natural indentation and model adjustment . . . . . . 874.3.2 Life calculation . . . . . . . . . . . . . . . . . . . . . . 92

4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

5 General conclusion 95

A Transient EHL solver implementation 111

B Dry contact analytical model 117

C Line contact stress 123

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1

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2 CONTENTS

Notation

α pressure viscosity index [Pa−1]α dimensionless viscosity index [.] = αpH

δ correction [.]∆P dimensionless additional pressure amplitude [.]S∆P initial slope of the dimensionless

additional pressure [.] =d∆P (D/Φ)

d(D/Φ)

D/Φ→0

η viscosity [Pa.s]η dimensionless viscosity [.] = η/η0

η0 viscosity at ambient pressure [Pa.s]κ ellipticity [.] = b/aλ roughness wavelength [m]

λ parameter of Reynolds equation [.]= 12umη0R

2x/(b3pH)

∇ dimensionless wavelength parameter [.]

= λ/b ·√

M/L∇Φ dimensionless indent parameter [.]

∝ Φ ·√

M/Lν function of the decay coefficient K [.]ν1, ν2 Poisson coefficient for body 1 and 2 [.]ω adjustment coefficient [.]ω under relaxation coefficient [.]Ω integration domain [.]φ indent diameter [m]Φ dimensionless indent diameter [.] = φ/bρ radius of curvature [m]ρ density [kg/m3]ρ0 density at ambient pressure [kg/m3]ρ dimensionless density [.] = ρ/ρ0

σ′ stress [Pa]σ dimensionless stress [.] = σ′/pH

τ stress criterion for the stress risk integral [Pa]ϑ failure probability [.]ξ dimensionless coefficient in Reynolds equation [.]ζ function of the decay coefficient K [.]

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CONTENTS 3

a Hertzian contact half width along y (major axis ellipse) [m]A line relaxation matrixb Hertzian contact half width along x (minor axis ellipse) [m]c stress exponent [.]d indent depth [m]D dimensionless indent depth [.] = dRx/b2

e Weibull slope [.]E1, E2 Young modulus for body 1 and 2 [Pa]E′ equivalent Young modulus [Pa]E elliptical integral of the second kind [.]f right member [.]F ratio between initial and deformed indent depth [.] = Dd/Di

F failure probability [.]h depth exponent [.]h gap height [m]hX , hY dimensionless mesh size [.]hT dimensionless time step [.]H dimensionless gap height [.] = hRx/b2

H0 dimensionless mutual approach [.]i, j index in space [.]I risk integral

I dimensionless risk integral [.] = (I/(b3 pcH))1/c

IHh , Ih

H restriction and interpolation operatorsIIh

H , IIHh restriction and interpolation operators (MLMI)

k index in time [.]K decay coefficient [.]K elastic kernel [.]L differential operator

L Moes parameter [.] = α E′ · ((2 η0 um)/(E′ Rx))1/4

m adjustment coefficient [.]

M Moes parameter [.] = w/(E′ R2x) · ((2 η0 um)/(E′ Rx))−3/4

n adjustment coefficient [.]nX , nY number of points along X and Y [.]p pressure [Pa]P dimensionless pressure [.] = p/pH

pH maximum Hertzian pressure [Pa]PREF dimensionless reference pressure [.]qK function of the decay coefficient K [.]qT∞

function of the dimensionless infinite life [.]r residual term [.]

r radius [m] =√

x2 + y2

rh shoulder top radius [m]rhX , rhY dimensionless inverse mesh size [.] = 1/hX , = 1/hY

rhT dimensionless inverse time step [.] = 1/hT

R dimensionless radius [.]Rx1, Rx2 radii of curvature along X for body 1 and 2 [m]Ry1, Ry2 radii of curvature along Y for body 1 and 2 [m]

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4 CONTENTS

Rx, Ry reduced radii of curvature [m]R roughness or indent [m]R dimensionless roughness or indent [.]s adjustment coefficient [.]sh shoulder height [m]t time [s]T dimensionless time [.] = tum/bT dimensionless stress criterion for the stress risk integral [.] = τ/pH

u speed [m/s]u unknownv numerical error [.]w load [N]x, y, z coordinates [m]X, Y, Z dimensionless coordinates [.] = x/b,= y/b, = z/bXa, Xb dimensionless domain boundaries along X [.]Ya, Yb dimensionless domain boundaries along Y [.]

©DRY dry contact©EHL EHL contact©V M Von Mises©PA Papadopoulos

© not converged© corrected

© coarse grid variable© discrete

©h fine grid©H coarse grid©b hole©d deformed©h hydrostatic©h shoulder©H Hertzian©i initial©GS Gauss-Seidel©JA Jacobi©PT pseudo-transient©∞ infinite life limit©m mean©FRONT front indent shoulder (first in the contact)©LAT lateral indent shoulder©REAR rear indent shoulder©CENT indent center©D dented©S smooth

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Chapter 1

General introduction

Tribology is literally the study of friction between two contacting surfaces.By extension, tribology treats the associated problems of lubrication andsurface deterioration (wear, fatigue. . . ). It includes phenomena that occurin everyday life and is implicated in a wide range of mechanisms. Actu-ally, every contact is subjected to friction (desirable or not) and surfacedeterioration. The understanding of the underlying mechanisms requires acombination of knowledge from various fields (physics, chemistry, materialscience, mechanical engineering, mathematics. . . ). This work is restrictedto the domain of elastohydrodynamic lubrication (EHL).

EHL contacts are encountered when a lubricant (frequently oil or grease)separates the two contacting bodies and the contact pressure is so high thatboth surfaces are elastically deformed. The lubricant limits friction andwear. The lubricant film formed between the two surfaces is very thin,typically thinner than 1 µm. The high contact pressure leads to a radicalchange in the lubricant behavior which becomes almost solid (piezoviscouseffect). These two aspects characterize EHL contacts: important surfacedeformation and important piezoviscous effects.

This kind of contact is found in various applications: cam tappet, gears,rolling element bearings. . . This work is mainly dedicated to the study ofcontacts in rolling element bearings.

5

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6 CHAPTER 1. GENERAL INTRODUCTION

The function of a rolling element bearing is to enable a relative rotationbetween two elements with precision and minimum frictional power losses.Rolling element bearings are constituted by two concentric rings, a set ofrollers and a cage, see figure 1.1. The two rings rotate at different speeds.The rollers adapt this velocity difference and transmit the load minimizingthe friction. Decreasing friction has been a crucial aspect1. The load istransmitted between the rings and the rollers by pressures created on thecontact areas. The cage separates the rollers.

Figure 1.1: Tapered roller bearing

The system of equations describing the EHL problem is complicated.Some advanced numerical techniques have to be used. The chapter 2 givesan overview of the equations and of the numerical techniques used. Figure1.3 represents a typical EHL pressure and film thickness distribution alongthe rolling direction. The lubricant is dragged into the contact sticking tothe surfaces. The contact can be divided in three zones: the inlet, wherethe pressure builds up but the viscosity remains sufficiently small to enableviscous flow; the high pressure region, where the viscosity is so high that thelubricant is almost rigid; and the outlet, with the characteristic constriction.

1Looking at figure 1.2, the reader probably wonders what the link is between rollingelement bearings and ten guys pulling a big stone lain on tree trunks. Well, it emergesfrom a literature survey starting 15 years ago with comics, that the figure 1.2 representsone of the oldest rolling-element-bearing-like documented applications (Iron Age).

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1.1. POLLUTION 7

Figure 1.2: Struggle against friction

1.1 Pollution

The contact life prediction is a major industrial issue. A large proportion ofreal mechanisms are subjected to contact deteriorations which affect theiroverall performance. Contact lives are important regarding economical as-pects; the user safety is also sometimes involved. Increased understandingof the contact behavior has enabled improvement of the reliability and theefficiency of industrial or individual mechanisms. In most engineering appli-cations, contact conditions become increasingly severe (higher loads, lowerviscosities. . . ). Some contradictory interests are involved: on one hand in-creasingly severe operating conditions and power loss diminution, and onthe other, reliability and security improvements. To achieve the best com-promise in design, the theoretical models have to be continuously improved.

Manufacturing processes have evolved during the last decades. A bearingwhich would have failed forty years ago because of very dangerous inclusionsin the material, has more chance nowadays to fail because of surface defects(roughness. . . ). Progress in material manufacturing has resulted in an in-

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8 CHAPTER 1. GENERAL INTRODUCTION

−2 −1 0 10

0.2

0.4

0.6

0.8

1

X

pressurefilm thickness

Inlet OutletHigh pressure

Figure 1.3: Typical EHL pressure distribution and film thickness along therolling direction

crease in the relative importance of other failure sources, and especially thelubricant pollution problems. Ai [5] reports that almost 75% of prematurebearing failures are related to pollution problems.

Lubricants contain particles. These particles are both initially present inthe lubricant (due to manufacturing process, storage. . . ); initially present inmechanisms (manufacturing residual particles. . . ); generated during runningin, wear and corrosion; or come from external sources (incomplete sealingand environment pollution, pollution during maintenance). The concentra-tion, size and material properties of these particles can vary substantially,see Ai [5]. Ville [119] gives a simple example of the initial pollution of thesame fresh oil which has been bought in a one liter packaging or two hun-dred liter packaging. The number of particles is multiplied by more than afactor four. Moreover, manufacturing residual particle specifications becomemore and more important. A strict process cleanliness is required to avoidparticles exceeding 400 µm for series production bearings.

Very different particles can be encountered in real lubricants. Boththe particle shape and properties can vary substantially. Table 1.1 brieflypresents different particles encountered in real devices and their usual con-sequences. Figures 1.4 and 1.5 show a picture of ductile metallic spheresand fragile sand particles from Ville [119]. The bearing surface indentationcaused by ductile metallic particles is the main object of this work.

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1.1. POLLUTION 9

Nature Origins Frequency Indentation

metallic–ductile

machining,assembling,wear, damag-ing

very important fornew parts, importantduring running-in

large and shal-low indents (lam-inated particles)

carbides–tough

machining moderate and lim-ited to new parts

sharp and smallindents (difficultto predict)

mineral–fragile

foundry sand,environment

limited to very im-portant dependingon the environment

very small indents(particle fracture)

Table 1.1: Particle origins and consequences on indentation [119]

Figure 1.4: M50 particles [119](distribution 0 − 100 µm)

Figure 1.5: SiC particles [119](mean size 45 µm)

These particles can be much larger than the film thickness. Typically,the particles can reach a few hundred micrometers, whereas the usual filmthickness is less than one micrometer. When a particle enters the contact, itis squashed between the surfaces and plastically deforms them. The remain-ing deformed surface geometry (a hole surrounded by shoulders) is called anindent. The indents perturb the contact pressure distribution. These pres-sure perturbations cause additional stresses in the material; these additionalstresses increase the fatigue failure risk (crack and spalling). Figure 1.6 rep-resents a fatigue failure initiated in the vicinity of an artificial indent.

The magic solution to the pollution problem does not exist. Several pos-sibilities are available: material development / thermochemical treatments,filters. . . However, even if all these approaches can bring great improvementsin mechanism life, the failure prediction remains a question of great inter-

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10 CHAPTER 1. GENERAL INTRODUCTION

Figure 1.6: Spalling due to an artificial indent [82]

est. For example, a classical filtering size in the automobile industry is 40 m.Webster et al. [124] still report a ratio of 7 between the characteristic L10life with a 40 µm filter and a 3 µm filter on a ball bearing life test bench.The use of very small filtering size is questionable because of the chokingrisk and the pressure drop. In this study, the aim is to predict indentedcontact life assuming: that the surface indentation is known, and that thematerial behavior is accounted for by an adjustment to experimental tests.

1.2 Modeling the pollution impact

1.2.1 Lubricant pollution and surface indentation

An important aspect is the link between a specific lubricant pollution andthe indentation which will be observed on the surfaces. Several studies havebeen conducted concerning the link between the lubricant pollution andthe surface indentation. On carefully designed experimental devices, understrictly controlled conditions, some clear conclusions have been obtained. Adirect link between the particle concentration and the indent surface densityhas been established by Ville and Nelias [122].

However the next step, which consists in applying these results to realcases, is not trivial at all. For example, the study mentioned above con-tradicted results obtained by Dwyer-Joyce [37] who related a ratio betweenparticle concentration and surface indentation density until 10 000 timeshigher.

Moreover, the relation between the indent geometry and the particleshape, the particle nature, the material behavior and the operating condi-tions is not clearly known. For example, the elastoplastic model which isused, strongly influences the calculated residual stresses. The prediction of

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1.2. MODELING THE POLLUTION IMPACT 11

the surface indentation from oil contamination in a real device is difficult.To be reliable, such a prediction in real cases would imply to know the con-centration and the nature of particles which will enter the contact. It is notonly strongly dependent of the particular bearing technology, but also fromthe polluted lubricant supply. Indeed, studies mentioned above, on the linkbetween particle concentration and indentation density, have been obtainedtaking many precautions concerning the lubricant supply and the mixing ofparticles in the lubricant. Let us imagine the complexity of an automobilegear box lubricant repartition which is often ensured by oil projection andcarter grooves. . .

Moreover, the way the particles will indent the surfaces depends on manyparameters such as the material properties or the contact conditions. Forexample, Ville [119] shows the major impact of particle material (shallow andwide indent for ductile particle versus sharper and smaller indent for toughcarbide particle). In this work, the surface indentation is supposed to beknown. For example, from a real mechanism or from predenting process aspresented by Nixon and Zantopoulos [93] or Girodin et al. [41], the surfacescan be analyzed using topography measures to quantify the indentationgeometry and density. With such data, the contact lives can be estimatedusing the model developed in this work.

Figure 1.7 gives a glimpse of where this work is situated in the entirecontact pollution problem. The example of an automobile gear box is taken.The surface indentation is assumed to be known. The contact condition andthe indent geometry are numerically modeled to obtain the pressure andstress distribution. A life model is then used to predict the life from thestress distribution.

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12 CHAPTER 1. GENERAL INTRODUCTION

Figure 1.7: Pollution modeling in a real device

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1.2. MODELING THE POLLUTION IMPACT 13

1.2.2 Life prediction

Prediction of rolling contact fatigue is complex. Many physical phenomenaare coupled at different scales. This work is focused on the macroscopicpressure and the stress distribution in indented contacts.

One of the major problems is to account for the cumulative damage.At each loading cycle, the material is submitted to stress variations. Apart of the energy provided to the material is converted in microstructuralchanges (dislocation movements, grain interactions, microstructural phasechanges. . . ) and other dissipative phenomena (heat generation, oil degra-dation. . . ). This microscopic view has a physical basis. However, even withvery simple loading cycles, the microscopic material behavior prediction isextremely difficult. Steel is an extremely complex material because of its het-erogeneity, several phases coexists, various defects are randomly distributed,and the microstructure is not continuous at all at the grain scale and evolveswith time and load cycles. Figure 1.8 represents the microstructure of a casecarburized quenched 16NiCrMo13 steel (residual austenite and martensitelath). However, a homogeneous and isotropic material will be considered inthis work, the phenomena mentioned above are more or less averaged and,the variability will be accounted for by a statistical model see chapter 4.

Figure 1.8: Microstructure of a case carburized quenched 16NiCrMo13 steel[108]

Important variations are observed in contact life. Some variations can beattributed to complicated loading cycles and stress distributions, materialproperties, manufacturing process influence. . .With so many variables andcoupled effects some restrictive assumptions have to be made. Neglectingsome variation causes and considering intrinsically random material proper-ties, implies to accept a large statistical dispersion in experimental results.The models used are idealized problems, and only attempt to highlight someof the principal physical trends and several order of magnitude variations.

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14 CHAPTER 1. GENERAL INTRODUCTION

Actually, the kind of theoretical models developed in this work have to becorrelated with experimental validations.

The statistical dispersion is unavoidable for many reasons: the prin-cipal one is that the material defects such as inclusions are intrinsicallyrandomly distributed and influenced by many parameters; the bearing ge-ometry varies also; the operating conditions are not completely controlled;the defect shapes (such as indentations) are submitted to random distribu-tions. . . All these aspects have to be kept in mind when looking at all thefuture considerations. Conclusions derived from models presented in thiswork have to be sufficiently significant to be experimentally correlated. Itis irrelevant to highlight a 5% difference in the calculated life. Indeed, the90% confidence band of a standard life test campaigns2 is wide. If one re-peats the standard life test, one will obtain several characteristic lives. Thelife interval which regroups 90% of the characteristic lives is spread roughlyfrom half to twice the median value.

Finally, the intermediate step of the pressure analysis will be studiedhere for two main reasons: first, understanding the role of different parame-ters is easier when proceeding step-by-step (pressure distribution, subsurfacestress, life reduction). Moreover, the pressure results do not depend on thefatigue life model. The calculated pressure distributions, which are alreadyobtained using several assumptions, are certainly less questionable than fa-tigue lives and can be more easily experimentally validated.

1.3 Objectives

Different bearing technologies exist. The contact conditions can changeenormously. So, a life prediction tool as accurate as possible is of great in-terest to avoid time consuming and expensive test campaigns. Moreover, forpractical (and cost) reasons, the products are tested on a certain range of lifeand operating conditions. The extrapolation from these experimental datato a larger range of operating conditions is done using a theoretical model.The theoretical models developed here use three major steps: the devel-opment or the extension of numerical tools (elliptical dry contact pressure,transient elliptical EHL contact pressure, contact stress. . . ), the analysis ofthe calculated results and the development of models.

This work focuses on three points considering the pressure and stressdistribution: the ellipticity influence, the indent geometry including theshoulder role and the operating conditions. As explained previously, thefinal step of the life prediction is strongly dependent on material properties.So, only general trends will be presented.

• Chapter 1 gives the context and some limitations of this study. It

2six failures, in first in four tests

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1.3. OBJECTIVES 15

restricts the question of the life prediction of a bearing in a real de-vice under polluted environment, to the simpler3 question: knowingthe indentation distribution (density and geometry), what will be thepressure and the subsurface stress distributions as a function of theoperating conditions?

• Chapter 2 presents the dry contact and EHL contact models. Itpresents the main equations and a brief overview of the numericaltechniques.

• Chapter 3 studies the contact ellipticity influence. It presents someconclusions about the contact ellipticity influence on the pressure dis-tribution. The dry and EHL indented contact pressure distributionare compared. A dry contact pressure distribution model is presented.The lubrication condition influence is also detailed.

• Chapter 4 presents the stress distribution and life prediction results.The life model is presented.

• Chapter 5 summarizes this study and presents the future work.

3nevertheless sufficiently complex

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16 CHAPTER 1. GENERAL INTRODUCTION

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Chapter 2

Modeling and numerical

methods

2.1 Introduction

This work is a numerical study of indented contacts. The main equationsare presented below. An effort will be made to point out the advantagesand the limits of the models. The goal of this study is to predict indentedcontact behavior for a wide range of geometries and operating conditions.It means that hundreds of calculations are required. The other side of thecoin is that some restrictive choices have to be made to be able to obtainthese results in a reasonable time.

The EHL problem is governed by a complex set of equations. It requiresa very detailed solution. A fine mesh implies a large number of unknownsand the solution time can become unacceptable unless advanced numericaltechniques are used. The numerical techniques should enable to deal withthe numerical difficulties of the set of equations. Moreover, obtaining asolution is not sufficient, an idea of the numerical error should be knownas well. Below, an overview of the following aspects will be given: drycontact pressure, EHL contact pressure and subsurface stress distributioncalculation.

17

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18 CHAPTER 2. MODELING AND NUMERICAL METHODS

2.2 Dry contact

The contact pressure distribution under dry contact conditions has beenstudied by Hertz [53]. The dry contact solution referred as Hertzian contactsolution is considered as a limit case of lubricated contacts. Actually, whenfilms become very thin and hydrodynamic effects decrease (high loads, lowspeeds, and low viscosities), the Hertzian pressure distribution is the asymp-totic solution.

The Hertzian solution is the extension of the work of Boussinesq [13] whohas formulated several assumptions: the bodies are semi-infinite half space(the contact zone is small compared with the body sizes); the bodies areelastic, homogeneous and isotropic; the stresses are normal to the surface.

The equations below will be presented twice: initial equations and di-mensionless equations. The equations are solved using dimensionless vari-ables for two main reasons: first, the dimensionless parameters enable toobtain variables of the order of magnitude of 1 which is crucial for numericalprecision; second, they enable to use the opportunity of similarity groups.

2.2.1 Gap height equation

Ry2

Ry1

Rx1

Rx2

h0

E1

E2

x

E’

E’

yh0

RxRy

O

Figure 2.1: Contacting body geometries – real geometry (left) and equivalentgeometry (right) [28]

The contacting body geometries can be well approximated by paraboloidsbecause the contacting body radii of curvature are large in comparison withthe contact dimensions. Since the surfaces are approximated by paraboloids,the equivalent radii of curvature Rx and Ry can be defined for a non-conforming contact by equation 2.1. Figure 2.1 represents, on the left the“real” geometry, and on the right the equivalent geometry.

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2.2. DRY CONTACT 19

1

Rx=

1

Rx1+

1

Rx2

1

Ry=

1

Ry1+

1

Ry2(2.1)

The gap height equation represents the separation between the two bod-ies. When the two bodies come into contact, a pressure is generated. Thepressure deforms the initial undeformed surfaces. This deformation is calcu-lated using the deformation integral which is function of the contact pressureover the domain Ω.

h (x, y, t) = h0 +x2

2Rx+

y2

2Ry+ R (x, y, t) +

2

πE′

∫∫

Ω

p (x′, y′, t) dx′ dy′√

(x − x′)2 + (y − y′)2(2.2)

The term h0 is the mutual approach of two remote points in the solids.The indent geometry will be introduced through R. The equivalent Youngmodulus corresponds to:

2

E′=

1 − ν21

E1+

1 − ν22

E2(2.3)

The dimensionless gap height equation reads:

H (X, Y, T ) = H0 +X2

2+

Rx

Ry

Y 2

2+ R (X, Y, T ) +

1 + Rx/Ry

2πE

∫∫

Ω

P (X ′, Y ′, T ) dX ′ dY ′

(X − X ′)2 + (Y − Y ′)2(2.4)

κ

(

Rx

Ry

)

≈ 1

1 +

ln

(

16/Rx

Ry

)

2Rx

Ry

−√

ln 4 + 0.16 lnRx

Ry

(2.5)

E(

1 − κ2)

=

∫ π

2

0

1 − (1 − κ)2 sin2 ψ dψ (2.6)

E(

1 − κ2)

≈ π

2κ2

(

1 +2

(

1 − κ2)

πκ2− 0.125 lnκ2

)

(2.7)

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20 CHAPTER 2. MODELING AND NUMERICAL METHODS

The elliptical integral approximation E in equation 2.7 is from Ham-rock and Brewe [52] and Moes [88]. For circular contacts, Rx/Ry = 1 ,the elliptical integral E tends asymptotically to π/2 while the ellipticity κbecomes 1.

E (0) =π

2(2.8)

2.2.2 Force balance equation

The force balance equation represents the load equilibrium. The appliedload is compensated by the pressure which is generated on the surfaces. Theforce balance is considered in a quasi-static approximation. It means thatno inertia is considered, and the load balance is exactly and immediatelyverified.

∫∫

Ωp (x, y, t) dx dy = w · f(t) (2.9)

The dimensionless force balance equation reads:

∫∫

ΩP (X, Y, T ) dX dY =

3κ· f(t) (2.10)

2.3 EHL contact

The fundamental equation of thin film lubrication has been established byReynolds [98]. This equation is derived from the Navier-Stokes fluid dy-namics equations using some simplifications and assumptions. The majorshortcuts are: a very thin lubricant film thickness in comparison with thecontact dimensions, a continuous fluid which sticks to the surfaces, a lami-nar flow, a constant pressure through the film thickness, volume and surfaceforces are neglected in comparison with viscous forces. The Reynolds equa-tion describes the mass flow conservation of a viscous fluid flow betweentwo parallel surfaces. It links the pressure to the contact geometry and thelubricant properties.

A major step forward in EHL understanding was in the work of Ertel [38]the combination of Hertzian solution and a piezoviscous lubricant behaviorfrom the work of Barus [8]. The first numerical solution of an EHL con-tact is proposed by Petrusevich [96] who solved simultaneously the elasticdeformation and the Reynolds equation. The smooth contact film thicknessstudy was published by Dowson and Higginson [32] for line contacts; andlater by Hamrock and Dowson for elliptical contacts [49, 50, 51].

The experimental measurement were improved with the interferometrictechniques. The gap height between the two surfaces is reconstituted fromthe interference pattern of light rays see Gohar and Cameron [43] and Foord

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2.3. EHL CONTACT 21

et al. [40]. Thin film measurements (5 nm) were made possible by Johnstonet al. [68].

The EHL contact equations are complex. Three equations are simulta-neously solved: the two previous dry contact equations (gap height equation2.4 and force balance equation 2.10) and the Reynolds equation. This set ofequations presents some pathological behaviors:

• A strong coupling between these three equations: for example, a mod-ification of the mutual approach from the force balance equation in-duces a global variation of the film thickness and so, of the pressuredistribution.

• Very non-linear behavior of the viscosity with pressure.

• Nature change of the Reynolds equation between low and high pressureregions (from elliptical to hyperbolic).

• The integral part of the elastic deformation equation: a pressure vari-ation in a single point affects the entire deformed geometry.

A major step forward in the solution of these equations is due to thedevelopment of multigrid techniques by Brandt [14, 15]. These techniqueswere applied to the EHL problem by Lubrecht [79, 80, 81]. To speed upthe elastic deformation calculation, Brandt and Lubrecht [16] developedthe multigrid multi-integration scheme (MLMI). Venner [109] optimized therelaxation process of the multigrid solver. The improvement concerning thestability and the convergence speed enabled to solve transient problems.

The transient problem has been studied for various geometrical patternsand especially harmonic roughness, see Lubrecht [86], Venner [110, 112,111, 113, 114]. A different approach is developed by Greenwood [44, 45]and Morales-Espejel [89]. Hooke [54, 55, 56] proposed several studies wherethe importance of the ratio between the geometry wavelength and the in-let length is highlighted. Ai [1] calculates the effect of a moving indenttraversing an EHL contact.

The experimental validation was done in parallel of the developmentof high speed camera and interferometric measurement Kaneta et al [71,72], Wedeven and Cusano [126]. The lubricant behavior is at center ofattention of latest developments, see Chapkov [18, 19] and Hooke [61, 62].An important number of publications are also related to the microgeometryoptimization as a function of the operating conditions by surface texturing(especially on coatings), both on numerical and experimental aspects [123],[90]. The lack of lubricant referred as starved contacts is also numericallystudied for example by Chevalier [22] and Damiens [28].

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22 CHAPTER 2. MODELING AND NUMERICAL METHODS

Figure 2.2: Comparison of experimental (left) and numerical (right) resultsof a transverse ridge under pure-rolling, experimental result by Kaneta [71]and numerical result by Lubrecht [110]

2.3.1 Lubricant

An important aspect of the EHL lubrication is the lubricant behavior. Theoperating conditions are extreme in terms of pressure (several GPa) andsometimes shear rate. Some very sophisticated lubricant behavior modelexist. For example, the lubricant viscosity-pressure relationship η is oftenvery dependent on the shear rate. However, for the range of small slide toroll ratio (encountered in rolling element bearings < 5%), a simple model ofNewtonian behavior is a reasonable simplification. Moreover, with a zero orsmall sliding speed, the thermal effects are expected to be small and will benegliged.

The Roelands viscosity [99] is used:

η (P ) =η

η0= e

αprz

−1 +

(

1 +Pph

pr

)

z

αpr

z= ln η0 + 9.67 (2.11)

pr = 1.96 108

The fluid compressibility appears to have a small effect on the indentedcontact behavior. The simplest and most common density relation, whichwas proposed by Dowson and Higginson [33], is used:

ρ =ρ

ρ0=

0.59 · 109 + 1.34 · Pph

0.59 · 109 + Pph(2.12)

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2.4. SUBSURFACE STRESS DISTRIBUTION 23

2.3.2 Reynolds equation

The Reynolds equation is written for a constant speed, the transient termcomes from non smooth patterns traversing the contact:

∂x

(

ρh3

η

∂p

∂x

)

+∂

∂y

(

ρh3

η

∂p

∂y

)

= 12um∂ρh

∂x+ 12

∂ρh

∂t(2.13)

To make the Reynolds equation dimensionless, the parameter λ is definedin equation 2.15. The transient dimensionless Reynolds equation with aNewtonian fluid reads:

∂X

(

ρH3

ηλ

∂P

∂X

)

+∂

∂Y

(

ρH3

ηλ

∂P

∂Y

)

=∂ρH

∂X+

∂ρH

∂T(2.14)

λ =12 umη0R

2x

b3ph(2.15)

2.4 Subsurface stress distribution

The contact elastic tensor (CET) is defined as the elastic stress distributionassociated with the deformation in the contact. One can approximate theCET in each point of the semi-infinite half space which is submitted to agiven pressure field on the top surface. The influence coefficient method isused to compute the stress integrals. The problem is solved by meshing theloaded region on the top surface as a finite union of squares. The kernels usedin those integrations are deduced from Boussinesq and Cerruti equations[13]. The CET is calculated assuming homogeneous and isotropic bodies,small deformations and linearity between deformations and stresses.

MLMI techniques are used to calculate the integrals of the elastic stressfield to ensure a reasonable calculation time. The scheme used is fromLubrecht [84] with appropriate kernels, see Kalker [69]. The domain has beenmeshed as shown in the figure 2.3 to ensure both a sufficient discretisationof the high gradient region (close to the surface) and a coarser meshingdeeper (to maintain reasonable calculation times). The contact pressuresand subsurface stresses are calculated assuming no friction at the surface.The principal stresses are calculated according to Fenner [39].

The residual stress distribution and the contact elastic stress distributiondue to indent pressure perturbations are considered separately. The residualstress is the stress distribution in the material without any loading obtainedafter running-in. The real stress state in the material during loading is thesum of the residual stress distribution and the CET.

To predict the residual stress field one needs several steps: initial residualstress of the bodies (manufacturing residual stress, thermochemical treat-ment residual stress, hoop stress), the elastoplastic behavior of the two

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24 CHAPTER 2. MODELING AND NUMERICAL METHODS

kf ⇔ Z

c

kf / 2

⇔ Zc

kf / 2

⇔ 2 Zc

Figure 2.3: Stress meshing

bodies and the particles, the particle geometry and nature, the indenta-tion residual stress, the running-in residual stress (geometry stabilization).Moreover, as said in section 2.5, the particle indentation modeling is still achallenging problem. The indentation conditions are generally not known.

Finally, the residual stress distribution will be omitted. Two main justifi-cations can be given. Firstly, it would be very costly considering the numberof parameters (and so, very time-consuming). For example, elasto-plasticcalculations would no longer remain dimensionless. . . Secondly, most of thefatigue criteria are based on the stress amplitude seen by the material. Theresidual stress is expected to be stabilized very quickly in comparison of thecontact life. With a fatigue criterion like the Dang Van criterion [29], theresidual stresses do not act on this stress amplitude because it is a constantstress tensor shift. The impact is only on the residual hydrostatic stress.The validity of this particular point for high compressive hydrostatic stressis contested see Desimone et al. [30].

2.5 Indent geometry

EHL film thickness is classically thinner than one micrometer. Pollutionparticles can reach few hundreds of micrometers. When these particles enterthe contact, they are squashed between the two surfaces. The particlesplastically deform the two surfaces and the remaining geometry is calledan indent. The indent geometry is difficult to predict. Several numericaland experimental studies treat the indent geometry aspect. These studies

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2.5. INDENT GEOMETRY 25

use both artificial indents (Rockwell indenter) or natural indents (pollutedcontact).

The understanding of the indent creation is interesting for two reasons:first, the pressure and stress perturbations induced by an indent traversing acontact depend obviously on its geometry; second, the indentation residualstress can be correctly evaluated.

2.5.1 Natural indentation observations

Figure 2.4: Ductile particle indent 3D mapping by Cogdell [23]

Figure 2.5: Ductile particle indent [119]

A comprehensive study of the indentation especially with ductile parti-cles was made by Ville [119]. Some of the results are reported and are linkedto some experimental observations in tapered roller bearings performed byCogdell at the Timken company metrology laboratory.

Ville [119] describes the major impact of the particle nature (ductile,fragile, tough) and the operating conditions (especially sliding) on the in-dentation. For example, brittle particles (sand) explode in the contact inlet

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26 CHAPTER 2. MODELING AND NUMERICAL METHODS

and produce numerous small indents; tough ceramic (carbide) particles pro-duce sharp indents and seem to be a harmful type of contaminant; themetallic – ductile spheres produce large shallow indents, see figure 2.4, 2.5.For pure-rolling conditions and ductile particles, these indents are roughlyaxysymmetrical. The shoulder height seems to be a little higher on the rearshoulder.

• The Hertzian pressure seems to have a small influence on the indentgeometry.

• The particle diameter has an important influence on the indent diam-eter, however a more moderate influence on the indent depth.

• The mean velocity has a moderate impact on the indent depth anddiameter. The difference concerning the indent depth seems to belinked with the central film thickness variation. Moreover, the indentshoulders are much higher with decreasing speed.

• The slide to roll ratio has a very important impact on the indentgeometry. The indent shape becomes very irregular and stretchedalong the sliding direction even for small slide to roll ration (6%).

Particle diameter (µm) 10-20 20-32 32-40 40-50

Indent diameter (µm) 20-30 30-40 40-80 60-100

Indent depth (µm) 1.5-1.7 1.7-1.8 1.8-3.0 2.0-3.0

Indent slope () 8.7-6.7 6.7-5.2 5.2-3.8 3.8-3.0

Table 2.1: Experimental observation of the indent diameter, depth and slopeas function of the particle diameter [119]

Table 2.1 represents the indent slope and diameter distribution as afunction of the particle size. These measures have been obtained by Ville[119] on a twin disc pollution bench. The particles are M50 spheres, theycross once the contact creating a unique indent. Pure-rolling conditionshave been used. The discs are made of 52100 steel with a similar hardnessas the particles. When the particle size increases, the indent diameters arelarger and the indent slopes are shallower. Observations of bearings whichhave run under controlled polluted environment show also that the sharpestindents are systematically the smallest.

For a given class of particle diameters, the variation of the indent geom-etry is relatively limited. Moreover, the results presented above have beenobtained ensuring that the particles traverse only once the contact. How-ever, when a particle traverses several times the contact, it creates each timea shallower indent. In a rolling element bearing, particles can be trappedin the bearing and traverse many contacts. It can be an explanation why

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2.5. INDENT GEOMETRY 27

numerous very shallow indents are measured on the surfaces of a bearingwhich runs under controlled polluted environment.

2.5.2 Indent creation modeling

Artificial indentations (Rockwell indenter) have been modeled in severalstudies. The indentation residual stress influence on contact life which isstudied by Ko and Ioannides [74], Lubrecht et al. [85], Xu et al. [129].The authors observe a moderate impact of the indentation residual stressdistribution, only Hamer et al. [48] predicts an important impact. For drycontacts or pure-rolling EHL contacts, the residual stresses have a beneficialeffect on calculated life. The predicted life can roughly be multiplied at themost by a factor four (for low loads, and almost constant for high loadsand/or very severe indents). At first sight, it could seem important but thereduction in life presented in these studies between smooth and indentedcontacts is about 103. The calculated indentation is strongly dependenton the material properties (elastoplastic behavior). The friction coefficientseems to have no influence on the indent hole, and a moderate influence onthe shoulder geometry, see Nelias et al. [92].

However, concerning natural indentation, the study conducted by Kanget al. [73] shows that the friction coefficient between the particle and thebodies seems to have an important effect on the dent geometry. Particleindentation is also studied by Sayles and Ioannides [103], and Xu et al. [129]who confirm the moderate impact on indented contact life of the indentationresidual stress.

2.5.3 Analytical description

Indents present a hole surrounded by shoulders. The indent geometry hasto be modeled as precisely as possible, but keeping a reasonable number ofparameters. Two problems are encountered: First, the indent geometry de-pends on the operating conditions (nature of particles, sliding, . . . ); second,the indent geometry evolves during the bearing life. When an indent is cre-ated, the initial indent geometry evolves during a few thousand overrollingcycles until the pressure perturbations around the indent induce only elasticdeformations. It leads to the stabilized geometry. It is assumed that onlyvery few cycles compared to the total life are sufficient to obtain the stabi-lized geometry, which will be subjected to high cycle fatigue, see Bhargavaet al. [9, 10], Hahn et al. [47], Coulon [24].

Two major approaches exist, on one hand, the use of the measured indentgeometry and on the other, the use of simplified geometries such as analyti-cal indent geometry. The use of measured geometries has two disadvantages.First, the quality of the solution is not easy to evaluate and second, generaltrends are difficult to extract. The analytical indent description used in this

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28 CHAPTER 2. MODELING AND NUMERICAL METHODS

work is obviously an important simplification, see figure 2.6. However, itenables a parametric study (reduced number of parameters). The continu-ous geometry facilitates the numerical accuracy. Finally, the analysis usesthe indent geometry used by Coulon [24] which uses three parameters, seeequation 2.16.

R = −De−K

R2

Φ2 cos

(

πR

Φ

)

(2.16)

These three parameters control the indent shape: the diameter Φ, thedepth D and the attenuation factor K for the shoulder shape. The analyticalfunction 2.16 is a good compromise between the number of parameters, thecorrect emulation of the indent geometry and good numerical properties.Real surface indents under polluted environment present so many differentgeometries (variability in particle nature, size, operating conditions, . . . )that for the moment, no clear observation justifies a more sophisticatedanalytical description.

−1 −0.5 0 0.5 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

R

Z

K=3

K=4

K=5

K=8

Figure 2.6: Indent profiles for different decay coefficients K from 3 to 8,D = 1, Φ = 1

2.6 Numerical techniques

Following the modeling, the continuous equations have to be solved. Thesecontinuous equations are too complex to find an analytical solution, so theequations are discretised and the discrete solution computed in nodes of amesh grid. The accuracy of this step is determined by the discretisationerror. The equations are solved using an iterative process. At each itera-tion, the variables converge to the solution. The difference between the lastiteration and the discrete solution is the numerical error.

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2.6. NUMERICAL TECHNIQUES 29

The first crucial point in a numerical study is to ensure that the solutionrepresents accurately the continuous equations. In other words, the errorwhich comes from the numerical techniques used to solve the continuousproblem, should be controlled. The solution trends should be caused by thephysical parameter variations, not by numerical artifacts. An example of anumerical artifact is clearly demonstrated by Venner [116] where artificialtrends for low speeds in EHL contacts appear as a result of poor accuracy.

The end of this chapter is a very quick overview of the basics of thenumerical techniques used in this work. A complete description and analysiscan be found in the literature especially Venner and Lubrecht [115]. Thiswork started with the implementation of the transient terms in the EHLmultigrid solver: the Gauss-Seidel line relaxation and the distributive Jacobiline relaxation. Validation of the numerical accuracy is also presented.

The solution of the EHL equations requires: first, a fine meshing whichimplies a large number of unknowns N ; second, numerical techniques whichcan overcome pathological behavior of these equations (especially in termsof stability); finally, efficiency in terms of computation time. The notion ofalgorithm complexity is at the heart of the problem1.

2.6.1 MLMI principles

The calculation of the deformation integral and of the stress integrals aremade using the technique of influence coefficients. The calculation of thedeformation integral and of the stress integrals corresponds to a convolutionbetween an integration kernel K (influence coefficients) and the pressuresPi,j . The kernel gives the influence of a ponctual pressure in every pointof the half-space domain. A naive implementation of these integrals gives acomplexity of N2 for the deformation integral. MLMI2 enables a complexityof N · ln(N) for the deformation integral.

The main idea consists in taking advantage of the kernel smoothness.The kernel involved here are asymptotically smooth kernels. It means thatthe kernel gradient can become very small far from the singularity. In thisconditions, the kernel can be interpolated with a negligible accuracy lost.The crucial point is to note that the smoothness assumption is only made onthe kernel, and not on the pressures. This subtlety is referred as anterpola-tion [115]. In the zones where the kernel is not smooth enough, a correction

1The answer of the increasing computer speed is deceptive. For example, a fine 2Ddiscretisation of 1024×1024 points corresponds to 220 unknowns. A classic algorithm witha complexity of N

2 requires 240 operations. An “efficient” algorithm with a complexityof N · ln(N) requires roughly 224 operations. If the efficient method solves the problemin one minute, it means that the classic algorithm will require about one month and half.The CPU speed increase is expected to reach its limit in 10 to 15 years. Anyway, in 15years, CPU should be roughly 210 faster. So don’t hold your breath, it means that youstill will have to wait more than one hour to obtain your result.

2MultiLevel-Multi-Integration

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30 CHAPTER 2. MODELING AND NUMERICAL METHODS

is used. The correction has a complexity cost, so the correction has to bemade only when a gain is obtained on the discretisation error.

The continuous integral is:

w(x) =

ΩK(x, y)u(y) dy (2.17)

A piecewise integration is used:

whi,j = hd

j

Khhi,j uh

j (2.18)

wHi,J = Hd

j

KhHi,J uH

J (2.19)

uHJ = 2−d[(IIh

H)T uh. ]J (2.20)

Then, the fine grid integral whi,j is approximated by the coarse grid inte-

gral wHi,J and a correction is applied around the singularity:

whi ≃ wH

I + hd∑

|j−i|6m

(Khhi,j − Khh

i,j )uhj (2.21)

Khhi,j = [(IIh

H)KhHi,. ]j (2.22)

The deformation kernel is calculated using the equations 2.23 and 2.24.The stress kernels can be found in [69].

Ki,j,k,l =2

π2

(

|Xp| asinhYp

Xp+ |Yp| asinh

Xp

Yp+ |Xm| asinh

Ym

Xm+ |Ym| asinh

Xm

Ym

−|Xm| asinhYp

Xm− |Yp| asinh

Xm

Yp− |Xp| asinh

Ym

Xp− |Ym| asinh

Xp

Ym

)

(2.23)

Xp = Xi − Xk +hX

2Yp = Xj − Xl +

hY

2

Xm = Xi − Xk − hX

2Ym = Xj − Xl −

hY

2(2.24)

2.6.2 Multigrid principles

The discrete problem is solved iteratively. Each iteration is called a re-laxation. A crucial aspect of the relaxation process used is that the highfrequencies of the error converge much faster than the low frequencies. It

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2.6. NUMERICAL TECHNIQUES 31

means that after few relaxations the high frequencies of the error have van-ished whereas the lower frequencies remain unchanged. The multigrid algo-rithm uses this characteristic. The algorithm works on several levels whichcorresponds to different mesh sizes. It is much more efficient for two rea-sons: first, obviously a coarser grid is synonym of less unknowns and asmaller amount of work per relaxation; second, the ratio between the errorwavelength and the mesh size decreases and a low frequency on the finegrid corresponds to a less smooth error on a coarse grid. It means that thecoarser levels will speed up the convergence of the low frequencies of theerror.

The EHL problem is non linear so the Full Approximation Scheme (FAS)has to be used.

FAS scheme

A general way of writing a differential equation is:

L 〈u〉 = f (2.25)

with:

• L – continuous differential operator

• u – continuous unknown

• f – continuous right hand side function (known)

After discretisation on a grid with mesh size h, the discrete equationreads:

Lh⟨

uh⟩

= fh (2.26)

with:

• Lh – discrete differential operator

• uh – discrete unknown

• fh – discrete right hand side function (known)

The resolution starts, the initial unknown uh is transformed after eachrelaxation in the incompletely converged unknown uh. The exact solutionof the discrete problem is uh. It enables to define the residual rh and thenumerical error vh.

rh = fh − Lh⟨

uh⟩

(2.27)

vh = uh − uh (2.28)

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32 CHAPTER 2. MODELING AND NUMERICAL METHODS

Substituting equations 2.27 and 2.28 in 2.26 gives:

Lh⟨

uh + vh⟩

= Lh⟨

uh⟩

+ rh (2.29)

The coarse grid H is now used to calculate a correction to apply to thefine level h. The restriction and interpolation operators are respectively IH

h

and IhH .

LH⟨

uH⟩

= fH

(2.30)

uH = IHh uh + vH (2.31)

fH

= LH⟨

IHh uh

+ IHh rh (2.32)

After a number of relaxations on the coarse grid, a good approximationuH to the coarse grid variable uH is found. The fine grid old approximationuh is then corrected according to:

uh = uh + IhH(uH − IH

h uh) (2.33)

This scheme is applied recursively on several grid levels see figure 2.7,typically on five levels from 256 grid points for the finest grid to 16 pointsfor the coarsest grid. The choice of intergrid transfer operators is not with-out consequences, see [115]. An efficient way to obtain a first guess of theunknown on the finest grid is to solve at lower cost the problem on coarsergrids and interpolate it recursively on finer grids. This scheme is referred asFull Multigrid Cycle (FMG), see figure 2.8.

lν1

Sw lν1

Sw lν1

Sw lν0¶7

lν2¶7

lν2¶7

lν2

h1 = 8h

h2 = 4h

h3 = 2h

h4 = h

Mesh size

1

2

3

4

Level

Figure 2.7: Example of a V (ν1, ν2)-cycle with 4 levels [115]

Line relaxation

As said previously, the efficiency of the relaxation process is strongly depen-dent on the error wavelength. It is detailed in the local mode analysis in[115]. A local relaxation scheme is used. The equation will be discretised

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2.6. NUMERICAL TECHNIQUES 33

ie¶7IIh

H

iSw i¶7

ieIIh

H¶7

iSw i

Sw i¶7i¶7

ie¶7IIh

H

iSw i

Sw iSw i¶7

i¶7i¶7

ie

1

2

3

4

Level

Figure 2.8: FMG with one V-cycle [115]

using finite differences. It means that the discrete equations in the grid pointi, j will only make appear few neighboring unknowns. As the equations arenon linear, a local linearisation is used. The current approximation uh

i,j in

the grid point i, j will be corrected in uhi,j by the correction δh

i,j :

uhi,j = uh

i,j + ωδhi,j (2.34)

δhi,j =

(

∂ Lh⟨

uh⟩

i,j

∂ uhi,j

)−1

uh=uh

rhi,j (2.35)

Relaxing every points of the grid, if the very last corrected values areused, a Gauss-Seidel relaxation is performed; if the corrected values areupdated only at the end of each grid relaxation, a Jacobi relaxation is per-formed.

The EHL equations present a strong coupling in the direction X. Insteadof solving the equations changing one unknown at the time, a line relaxationcan be performed. It means that the corrections of the locally linearisedproblems will be calculated on an entire line. An important gain in therelaxation efficiency is obtained. A smart choice of the important couplingterms enable to obtain an hexadiagonal linear problem with nX unknownswhich can be solved without impairing the overall complexity:

Ajδhj = rh

j (2.36)

with:

• Aj – hexadiagonal linear system matrix

• δhj – correction in all points i of the line j

• rhj – residual in all points i of the line j

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34 CHAPTER 2. MODELING AND NUMERICAL METHODS

It has been shown in section 2.6.1 that the deformation in one point is afunction of the pressures in all grid points through the influence coefficientmatrix. It means that a change in pressure in one grid point will affect all theother grid points through the deformation integral. It leads to importantnumerical difficulties (convergence and stability). To overcome this, thedistributive relaxation is used in the high pressure zone. The distributiverelaxation is based on the deformation kernel properties. The influence ofa local change in pressure on the deformation in the other grid points islimited by the distributive relaxation. When a correction is applied in onegrid point, an opposite smaller correction in the other direction is appliedto the neighboring points. Two schemes are employed, the Gauss-Seidelline relaxation for the low pressure zones and the distributive Jacobi linerelaxation for the high pressure zones.

rhj = fh

j− Lh

. . . ; uhj−1;

. . . ; uhi−1,j ; u

hi,j ; u

hi+1,j ; . . .

; uhj+1; . . .

(2.37)

rhj = fh

j− Lh

. . . ; uhj−1;

. . . ; uhi−1,j ; u

hi,j ; u

hi+1,j ; . . .

; uhj+1; . . .

(2.38)

Depending on which relaxation scheme is used, the correction matrix systemis constructed:

• Gauss-Seidel line relaxation:

Ajδhj = rh

j (2.39)

Aji,k =

∂ Lh⟨

uh⟩

i,j

∂ uhk,j

(2.40)

uhj = uh

j + ωGS δhj (2.41)

• Jacobi distributive line relaxation:

Ajδhj = rh

j (2.42)

Aji,k =

∂ Lh⟨

uh⟩

i,j

∂ uhk,j

− 1

4

(

∂ Lh⟨

uh⟩

i,j

∂ uhk+1,j

+

∂ Lh⟨

uh⟩

i,j

∂ uhk−1,j

+∂ Lh

uh⟩

i,j

∂ uhk,j+1

+∂ Lh

uh⟩

i,j

∂ uhk,j−1

)

(2.43)

uhi,j = uh

i,j + ωJA

(

δhi,j −

1

4

(

δhi+1,j+

δhi−1,j + δh

i,j+1 + δhi,j−1

))

(2.44)

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2.6. NUMERICAL TECHNIQUES 35

2.6.3 Transient calculation implementation

Transient calculations face some specific numerical difficulties. The directSU2 discretising of the transient term in finite differences leads to numer-ical damping, see Wijnant [127], Venner and Lubrecht [110], Venner andMorales-Espejel [114] and Venner et al. [114]. A more sophisticated NU2scheme is used. A detailed implementation of the Gauss-Seidel and theJacobi line relaxation is given using the NU2 scheme in appendix A.

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36 CHAPTER 2. MODELING AND NUMERICAL METHODS

2.6.4 Numerical accuracy

Some results about the numerical accuracy are presented below. The evo-lution of numerical errors as a function of the number of points is studied.Several curves represent the error in the pressure calculation, the stress cal-culation and the stress risk integral calculation.

When an indent crosses a contact, it creates a pressure perturbation(positive over the shoulders and negative in the hole). The additional pres-sure amplitudes are defined in figure 2.9 for a centered indent. For drycontacts, this pressure distribution is axysymmetric, this is no longer truefor EHL contacts.

−1.5 −1 −0.5 0 0.5 1 1.50

0.5

1

X

P(X

,0)

−1.5 −1 −0.5 0 0.5 1 1.50

0.5

1

Y

P(0

,Y)

∆PEHLFRONT∆PEHL

REAR

∆PEHLCENT∆PEHL

LAT

Figure 2.9: Additional pressure amplitude definition

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2.6. NUMERICAL TECHNIQUES 37

Dry contact

The domain is [−2 2] × [−2 2]. Figure 2.10 shows that the error in theadditional pressure amplitude for Φ = 0.25 is roughly twice the error forΦ = 0.5. A minimum of 16 points per diameter is required to ensure anaccuracy of 1% on the additional pressure amplitude. A similar conclusionwas obtained by Coulon [24].

101

102

103

104

10−2

10−1

100

101

102

nX

erro

r on

∆P

DR

Y (

%)

Φ = 0.25Φ = 0.5

Figure 2.10: Error in ∆PDRY for a dry contact as a function of nX . Indentgeometry Φ = 0.25 0.5, D/Φ = 2, K = 4

EHL contact

EHL indented contact calculations are transient. A steady-state problemwith a longitudinal groove is first used, see equation 2.45 and figures 2.11,2.12. Then, the error in the full transient problem is presented in figures2.13, 2.14. The domain is [−2.5 1.5] × [−2 2].

These plots show that a finer discretisation for the EHL problem isneeded. The error in the solution depends also on the operating condi-tions. The transient problem requires a minimum of 256 points to ensurean accuracy better than 10% on the additional pressure amplitude and of-ten enables an accuracy close to 2% for Φ = 0.5 and 4 − 5% for Φ = 0.25.A similar conclusion for harmonic wavelength was published by Lubrechtand Venner [86]. During the convergence process, an additional convergencecondition has been used. Usually, a condition on the mean residual is used.However, some locally large errors appeared close to the indent. A specificcondition on this maximum residual was added.

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38 CHAPTER 2. MODELING AND NUMERICAL METHODS

R = −De−K

Y 2

Φ2 cos

(

πY

Φ

)

(2.45)

101

102

103

10−2

10−1

100

101

102

nX

erro

r in

∆ P

EH

L (%

)

Φ = 0.125Φ = 0.25Φ = 0.5

Figure 2.11: Error in ∆PEHLLAT for a stationary EHL contact M = 200, L = 19

as a function of nX . Groove geometry Φ = 0.125 0.25 0.5, D/Φ = 1/4,K = 4

101

102

103

10−1

100

101

102

nX

erro

r in

∆ P

EH

L (%

)

Φ = 0.125Φ = 0.25Φ = 0.5

Figure 2.12: Error in ∆PEHLLAT for a stationary EHL contact M = 800, L = 8

as a function of nX . Groove geometry Φ = 0.125 0.25 0.5, D/Φ = 1/4,K = 4

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2.6. NUMERICAL TECHNIQUES 39

101

102

103

100

101

102

nX

error in ∆ PEHLREAR

(%)

Φ = 0.25Φ = 0.5

101

102

103

100

101

102

nX

error in ∆ PEHLLAT

(%)

Φ = 0.25Φ = 0.5

101

102

103

10−1

100

101

nX

error in ∆ PEHLCENT

(%)

Φ = 0.25Φ = 0.5

Figure 2.13: Error in ∆PEHLREAR ∆PEHL

LAT ∆PEHLCENT for a transient EHL contact

M = 200, L = 19 as a function of nX . Indent geometry Φ = 0.25 0.5,D/Φ = 1, K = 4

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40 CHAPTER 2. MODELING AND NUMERICAL METHODS

101

102

103

100

101

102

nX

error in ∆ PEHLREAR

(%)

Φ = 0.25Φ = 0.5

101

102

103

100

101

102

nX

error in ∆ PEHLLAT

(%)

Φ = 0.25Φ = 0.5

101

102

103

10−1

100

101

n

error in ∆ PEHLCENT

(%)

nX

Φ = 0.25Φ = 0.5

Figure 2.14: Error in ∆PEHLREAR ∆PEHL

LAT ∆PEHLCENT for a transient EHL contact

M = 800, L = 8 as a function of nX . Indent geometry Φ = 0.25 0.5,D/Φ = 1, K = 4

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2.6. NUMERICAL TECHNIQUES 41

Subsurface stress calculation

Pressure perturbations induce stress perturbations. High stress zones appearclose to the surface as shown in figure 2.15. The error in the maximum stressis presented in figure 2.16. Figure 2.17 represents the evolution of the errorin the stress risk integral3 ID with a constant mesh size in the subsurface.A 256× 256 top grid is required to ensure a sufficient accuracy of 1% on themaximum stress and the stress risk integral.

0.05 0.050.05

0.05

0.050.05

0.2

0.2

0.20.20.

3

0.30.3

X

Z

σVM in a vertical plane

X−2 0 2−4

−3

−2

−1

0

0.10.1 0.1

0.1

0.2

0.2

0.2

0.3

0.3

0.3

0.3

0.30.4

0.5

Z

0 0.5 1−0.8

−0.4

0

Figure 2.15: Example of a stress distribution in a vertical plane for a drycontact with a centered indent

3The stress risk integral definition will be detailed in chapter 4.

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42 CHAPTER 2. MODELING AND NUMERICAL METHODS

101

102

103

10−2

10−1

100

101

102

nX

erro

r on

σV

M (

%)

Φ = 0.25Φ = 0.5

Figure 2.16: Error in σV Mmax for a dry contact as a function of nX . Indent

geometry Φ = 0.25 0.5, D/Φ = 2, K = 4

101

102

103

10−2

10−1

100

101

nX

erro

r on

I D (

%)

Φ = 0.25Φ = 0.5

Figure 2.17: Error in ID for a dry contact as a function of nX . Indentgeometry Φ = 0.25 0.5, D/Φ = 2, K = 4

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2.7. CONCLUSION 43

2.7 Conclusion

This chapter has presented the models and the main assumptions for thecontact pressure and stress distribution calculation. Both the dry and theEHL models have been exposed. These two approaches will be used in thiswork. The simplifications have been emphasized. These simplifications areoften due to calculation time constraints and are the result of a compromisein the model. Classical transient Newtonian Reynolds assumptions are used.Pure-rolling conditions are assumed. Only pure elastic deformations areconsidered. The indent geometry is approximated by an analytical function.

A brief overview of the numerical solution techniques is given. Multigridtechniques are used for the deformation/stress integral calculations and forpressure calculation. The implementation details for the transient EHLsolver has been given. Finally, a rapid overview of the numerical accuracyis presented.

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44 CHAPTER 2. MODELING AND NUMERICAL METHODS

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Chapter 3

Dry and EHL pressure in

indented contacts

3.1 Introduction

The prediction of the indented contact pressure is an intermediate step in thelife prediction. The pressure distribution in dry and lubricated contacts isinteresting for several reasons. First, it is intuitively directly linked with thecontact severity. Second, the theoretical model is well established and filmthickness predictions are correlated by experimental observations. Finally,the understanding of the lubrication influence is more direct concerningpressure than stress or even risk integrals.

In this section, the indent geometry and the operating condition influenceare studied. The aim is to analyze the influence of different parameters(indent diameter, indent depth, shoulder geometry, contact ellipticity, speed,viscosity, load. . . ) over a certain range of operating conditions. It representshundreds of calculations which means hundreds of computing days. It alsotries to give a physical interpretation of the influence of these parameters.

Some results are presented using a reference pressure PREF . This refer-ence is used for reasons of confidentiality and has no physical meaning. Ithas a constant value and does not affect the different conclusions. The maininterest of these results is in the relative influence of the different parame-ters and in the physical interpretations rather than the numerical values ofpressure.

This chapter is divided in two main sections. First, dry contact as-sumptions are used. Then, lubricated contacts are studied. Each of thesetwo sections addresses three points: How does pressure evolve when an in-dent traverses the contact? What is the influence of the contact ellipticity?And finally, what are the important parameters concerning indented contactpressure and how can one predict the pressure perturbations induced by anindent?

45

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46CHAPTER 3. DRY AND EHL PRESSURE IN INDENTED CONTACTS

3.2 Dry contact pressure

The dry contact is used as a first approximation of the lubricated case. Itis seen as an asymptotic case of an EHL contact for low speed and viscosityand high load. There are also some practical advantages: the calculation israpid, stable and quasi static (the pressure distribution for an indent positiondoes not depend on previous solutions). The interpretation is easier and itprovides a reference for the future study of the lubrication influence.

Figure 3.1: 2D pressure distribution in an indented dry contact Φ = 0.5,K = 4

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3.2. DRY CONTACT PRESSURE 47

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1

1.5

2

X

PH

P∆P

∆ P

Figure 3.2: Additional pressure amplitude definition for dry contacts Φ =0.5, K = 4

3.2.1 Indent crossing

−2 0 2−1

0

1

X−2 0 2

−1

0

1

X

−2 0 2−1

0

1

X−2 0 2

−1

0

1

X

+/− PH

P∆P

Figure 3.3: Indent traversing a dry contact Φ = 0.5

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48CHAPTER 3. DRY AND EHL PRESSURE IN INDENTED CONTACTS

D

Φ= 1

D

Φ= 2

Φ = 0.5 < 1% < 1%Φ = 1 < 1% 2%

Table 3.1: Difference in the additional pressure amplitude during indent

crossing for dry contacts =∣

∣∆PDRYXD=0 − ∆PDRY

max(T )

∣ /∆PDRYmax(T ) as a function

of the indent diameter Φ and the indent slope D/Φ for K = 4

When an indent traverses a contact, it is deformed. A pressure dis-tribution is associated with this deformation. Figure 3.1 represents a 2Dpressure distribution in an indented contact. Figure 3.2 represents the pres-sure distribution and the additional pressure distribution along the rollingdirection X of an indented dry contact. Figure 3.3 represents the pressuredistribution along the rolling direction at different indent locations: froma smooth contact to a centered indent. If the pressure is divided into thesmooth pressure and the additional pressure distribution, one can observethat the additional pressure is almost constant for an indent traversing thecontact. The additional pressure is only shifted in the contact along therolling direction. One could have foreseen this result. It is convenient forseveral reasons. All the information about the additional pressure distri-bution during the entire crossing is obtained from a single centered indentcalculation, and the additional pressure amplitude for a centered indent isrepresentative of the entire crossing. It saves calculation time and memorybecause only one position is required.

This is true only for sufficiently small indents. It is quite obvious that ifthe indent is larger than the contact itself, it would be difficult to measurepressure over the shoulders outside the contact. More reasonably, the indentshould be sufficiently small to not perturb the force balance too much. Table3.1 presents the error in the maximum additional pressure amplitude duringthe indent crossing. For example, one can read that for a diameter Φ = 1 andan indent slope D/Φ = 2, the approximation error between the additionalpressure amplitude for a centered position and the maximum value duringthe indent crossing is 2%.

3.2.2 Ellipticity

The equations presented in chapter 2 are solved using dimensionless param-eters. These dimensionless parameters are especially useful because they al-low the complexity of the problem to be reduced. For example for a smoothcircular dry contact, the dimensionless equations enable one to obtain aunique solution which is applicable to every smooth circular dry contact

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3.2. DRY CONTACT PRESSURE 49

regardless1 of the radius of curvature, load, Hertzian pressure, contact halfwidth. . . The smooth circular dry contact pressure distribution is calculatedonce for all. Moreover, it demonstrates the level of understanding of theproblem which is reduced to its minimum expression.

The classical Hertzian dimensionless parameters are used. It means thatparameters describing indents depend on the contact conditions. For exam-ple a given “physical” indent will not have the same dimensionless depthand diameter if the contact half width changes.

P =p

pH

H =hRx

b2

X =x

b

Y =y

b

Elliptical contacts are commonly encountered in rolling element bearings.The ellipticity can vary substantially. A simple question2 appears: whenvarying the ellipticity, how do the pressure perturbations induced by a givenindent evolve? Actually, it depends on how one varies the ellipticity κ.There are three free parameters in the dry elliptical smooth contact. Severalarbitrary choices can be made; only one is presented here. The ellipticityvariation will be obtained with the contact half width b along the minor axis(rolling direction X) and the Hertzian pressure pH remaining unchanged.

Several reasons justify this choice: First, the ratio between the harmonicroughness wavelength and the contact half width b was identified by sev-eral authors as a key parameter for the deformation in lubricated contacts,see section 2.3. As the dry contact study is the basis of the lubricationstudy, the choice which was expected to simplify the lubrication study wasmade. Moreover, it maintains the dimensionless indent diameter constantwhen varying the ellipticity κ. Finally, the Hertzian pressure pH appearedintuitively to be an important parameter. A constant Hertzian pressurewhen varying the ellipticity simplifies the comparison of the dimensionlesspressures3.

The consequence is that the radius of curvature along the rolling direc-tion Rx is imposed and varies with the ellipticity κ. Equation 3.1 is obtainedfrom Hertzian equations; the left handside is constant when varying the el-lipticity.

1Actually, there are only two independent variables.2Because a simple question does not systematically give rise to a simple answer.3Certainly, they are made dimensionless with the same Hertzian pressure

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50CHAPTER 3. DRY AND EHL PRESSURE IN INDENTED CONTACTS

bE′

πpH=

4

π

E

1 +Rx

Ry

Rx (3.1)

The above equation is used in equation 2.4. For the two asymptotic casesof a circular and an infinitely long contact, one obtains:

circular contact:bE′

πpH= Rx

line contact:bE′

πpH=

4

πRx

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1

1.5

Y

Pκ=1.0

∆ Pκ=1.0

Pκ=0.1

∆ Pκ=0.1

Figure 3.4: Dry contact pressure distribution around an indent as a functionof the ellipticity Φ = 0.5

The result is that the additional pressure distribution remains almostconstant when varying the ellipticity. An example of the pressure distribu-tion around an indent when varying the ellipticity is given in figure 3.4, theadditional pressure distribution are perfectly superimposed. This is true upto a certain indent size. The force balance should not be perturbed sig-nificantly. Table 3.2 gives an idea of the validity of this observation. Forexample, one can see that the difference between the additional pressureamplitude for κ = 1 and κ = 0.1 for an indent with diameter Φ = 1 andindent slope D/Φ = 1 is 3%.

For sufficiently small indents, the ellipticity does not change the ad-ditional pressure amplitude. This is interesting for two reasons: First, iteliminates on one parameter of the parametric study. There is no need tocalculate all the different indent geometry pressures for several ellipticities.Second, wide elliptical calculations are time and memory consuming becausethe number of unknowns is proportional to 1/κ.

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3.2. DRY CONTACT PRESSURE 51

D

Φ= 1

D

Φ= 2

Φ = 0.5 < 1% < 1%Φ = 1.0 3% 7%

Table 3.2: Difference in the additional pressure amplitude for two ellipticitiesκ = 1.0 and κ = 0.1 for dry contacts =

∣∆PDRYκ=1.0 − ∆PDRY

κ=0.1

∣ /∆PDRYκ=1.00 as a

function of the indent diameter Φ and the indent slope D/Φ for K = 4

3.2.3 Analytical model for dry contacts

The previous section shows that the indent location question and the ellip-ticity variation question can be sidestepped. The next point is to highlightthe important parameters: the indent slope D/Φ and the shoulder geometryK, and to propose a predictive model for the pressure perturbations.

The analytical model developed below is based on the model developedby Coulon [24], Coulon et al. [25, 26] and Gupta et al. [46]. This model isbased on the elastic deformation of harmonic waviness studied by Johnson[67]. Coulon classified the indent in three categories: Zone A, indents arecompletely flattened in the contact; Zone B, the pressure becomes zero inthe indent hole and indents are no longer entirely flattened; Zone C, the loadis entirely carried by the shoulders. Coulon proposed an analytical model forthe additional pressure amplitude in zone A and C. This model is extendedto account for the shoulder influence in zone A and for the zero pressure inzone B, see figure 3.5.

−1 −0.5 0 0.5 10

0.5

1

1.5

2

X

zone Azone B

Figure 3.5: Zone A - B typical pressure distribution

Johnson shows that the additional pressure amplitude associated withthe complete deformation of a harmonic waviness is proportional to the

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52CHAPTER 3. DRY AND EHL PRESSURE IN INDENTED CONTACTS

slope. The same observation was made by Coulon, concerning indents inzone A. The additional pressure amplitude in zone A is linear in the indentslope D/Φ. The main assumption for the zone A model was that the loadnot carried by the hole because of the lack of material is totally compensatedby an annular additional pressure distribution referred to as the “not carriedload”, see figure 3.6.

Figure 3.6: Analytical model pressure distribution

However, analysis of the additional pressure shows that the additionalpressure amplitude ∆P depends on the shoulder geometry (controlled byK). Moreover, the “not carried load” is only partially compensated by theannular additional pressure. The load carried by the annular additionalpressure around the hole represents roughly 60% of the “not carried load”.The remaining load is spread over the entire contact area. Finally, the modelcan be extended to account for the partially flattened indents of zone B.

Table 3.3 presents the additional pressure amplitude evolution as a func-tion of the indent slope D/Φ, the shoulder geometry K and the indent di-ameter Φ. For example, an indent slope of 0.5, a shoulder parameter K = 3and an indent diameter Φ = 0.25 lead to an additional pressure amplitude∆P/PREF of 1.82. For this case, when varying the indent diameter Φ from0.25 up to 1.0, the additional pressure amplitude varies between 1.82 and1.72. Two conclusions can be drawn: first, the indent slope D/Φ and theshoulder geometry K have an important influence on the additional pressureamplitude ; second, the indent diameter Φ has a limited impact.

Extension of the zone A model

The aim is to include the shoulder influence in the analytical zone A model.Several simplifications are made, the geometry of the hole and the shoul-ders is approximated. The contribution of the pressure needed to flattenthe shoulders is modeled separately of the pressure needed to flatten the

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3.2. DRY CONTACT PRESSURE 53

D

Φ= 0.5

D

Φ= 1

D

Φ= 2

K = 3Φ = 0.25Φ = 0.50Φ = 1.00

1.821.821.72

3.273.252.96

5.835.784.83

K = 4Φ = 0.25Φ = 0.50Φ = 1.00

1.591.591.58

2.832.822.72

4.824.804.41

K = 8Φ = 0.25Φ = 0.50Φ = 1.00

1.261.261.30

2.052.052.09

3.103.103.16

Table 3.3: Evolution of the dry contact additional pressure amplitude∆PDRY /PREF as a function of the indent slope D/Φ, the shoulder geometryK and the indent diameter Φ

hole. The global additional pressure over the dent shoulder is divided in twocontributions:

∆Ptot = ∆Phole + ∆Pshoulder (3.2)

Figure 3.7: Shoulder approximation

The analytical approach of the shoulder influence uses a parabolic ap-proximation of the shoulder geometry, see figure 3.7. The shoulder radius ofcurvature and the shoulder height are used for the parabolic approximation.These two parameters are functions of D, Φ and K. The analytical addi-tional pressure amplitude is proportional to the indent slope with respect toa coefficient ν which is a function of the attenuation factor K, see equation3.3. A “circular” line contact of length 2πrh is assumed. Concerning the

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54CHAPTER 3. DRY AND EHL PRESSURE IN INDENTED CONTACTS

hole, the factor ζ in equation 3.4 is a function of K only, it depends also onthe simplified geometry used.

∆P ′shoulder ∝ D

Φν(K) (3.3)

∆P ′hole ∝

D

Φζ(K) (3.4)

Details about the calculation of ν and ζ are given in appendix B. Severalpossibilities have been tried to approximate the shoulder and hole geometry.However, they only lead to modify the proportionality coefficient in equa-tions 3.3 and 3.4. Two coefficients n and m are used to adjust the analyticalmodel to numerical results of zone A in equation 3.5.

Figure 3.8 represents the ratio between the additional pressure amplitudeand the indent slope as a function of the indent slope in zone A. Differentshoulder geometries are presented from K = 3 corresponding to high shoul-ders to K = 8 corresponding to flat shoulders. The model in dashed linescorrectly matches the full numerical results.

∆Ptot = n∆P ′hole + m∆P ′

shoulder (3.5)

0 0.025 0.05 0.075 0.1 0.1250

1.0

2.0

3.0

4.0

∆ P

/ (D

/Φ)

/ PR

EF

D/Φ

K = 3K = 4K = 5K = 8model

Figure 3.8: Ratio between the additional pressure amplitude and the indent

slope ∆PD/Φ in zone A (completely flattened indents) as a function of the

indent slope D/Φ and the decay coefficient K

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3.2. DRY CONTACT PRESSURE 55

Zone B model

Zone B corresponds to partially flattened indents. When the indent is suf-ficiently deep, it is not entirely flattened and a non contact zone remains inthe indent center where the pressure is zero. Figure 3.9 represents a typicalinitial and deformed indent geometry in zone B. The direct extension of thezone A model to zone B decreases progressively the hole contribution whenthe hole geometry is less and less flattened. The main idea of the zone Amodel is to compensate the “not carried load”. If the hole is no longer com-pletely flattened, the “not carried load” decreases. Keeping a sufficient holedeformation, it is expected that the shoulder influence remains constant.So, the transition from zone A to zone B is seen as a modulation of the holeadditional pressure contribution.

Figure 3.9: Example of deformed indent geometry in zone B (partially flat-tened indent)

A study of the ratio between deformed and initial indent depth Dd/Di

shows that it depends only on the indent slope, see figure 3.10. Two stepsare needed: first, linking the remaining hole depth with the indent slope;second adjusting the modulation of the hole additional pressure (coefficients and σ in equation 3.6). More details can be found in appendix B. Theadditional pressure reads:

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56CHAPTER 3. DRY AND EHL PRESSURE IN INDENTED CONTACTS

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

D/Φ

Dd/D

i

numericalmodel

ZO

NE

A

ZO

NE

B

Figure 3.10: Ratio between deformed and initial indent depth Dd/Di as afunction of the indent slope D/Φ

F = Dd/Di

∆Ptot = nD

Φζ(K) (1 − sFω) + m

D

Φν(K) (3.6)

Figure 3.11 represents the ratio between the additional pressure ampli-tude and the indent slope as a function of the indent slope in zone B. Agood correlation is obtained between the numerical results and the analyti-cal predictions (dashed lines). The difference increases with increasing slope.Indeed, with a remaining hole depth exceeding 80% of the indent depth, themodel is less accurate. Figure 3.12 recaps the good correlation obtainedbetween the full numerical calculations and the analytical model.

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3.2. DRY CONTACT PRESSURE 57

0 0.5 1 1.50

1.0

2.0

3.0

4.0

∆ P

/ (D

/Φ)

/ PR

EF

D/Φ

K = 3K = 4K = 5K = 8model

Figure 3.11: Ratio between the additional pressure amplitude and the indent

slope ∆PD/Φ in zone B (partially flattened indent) as a function of the indent

slope D/Φ and the decay coefficient K

0 2.5 5.00

2.5

5.0

∆ P

num

eric

al /

PR

EF

∆ Pmodel

/ PREF

Figure 3.12: Full numerical additional pressure versus analytical predictionfor Φ = 0.25 − 0.5, D/Φ = 0 − 2 and K = 3 − 8

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58CHAPTER 3. DRY AND EHL PRESSURE IN INDENTED CONTACTS

3.3 EHL contact pressure

The previous section studied the dry indented contact pressure. Below, thelubricated study is proposed.

Lubrecht et al. [83] presented a numerical work on the indented rollingbearing life. The contact was modeled by an EHL 1D line contact andthe indent by a transverse groove. The groove was chosen without anysharp corners and with significant shoulders emulating real indents. Theparametric study of the life prediction focused on the Hertzian pressure,the relative size of the indent related to the contact size, and the indentslope. The dry contact assumption was said to approximate closely the pure-rolling lubricated contacts. 2D point contacts were studied by Lubrecht etal. [85] and the authors concluded that previous studies using 1D geometrieswere not able to reproduce accurately 2D results. Ai and Cheng [1] studiedrolling sliding indented point contacts and highlighted the induced dentphenomenon described in section 3.3.4.

Figure 3.13 represents the pressure distribution in a pure-rolling EHLindented contact. The indent is centered. The additional pressure ampli-tudes are defined. The pressure distribution is more complex than in drycontacts. It is no longer symmetric and a plateau appears in the indent holewhereas a dry contact calculation would have given a zero pressure in theindent hole.

−1.5 −1 −0.5 0 0.5 1 1.50

0.5

1

X

P(X

,0)

−1.5 −1 −0.5 0 0.5 1 1.50

0.5

1

Y

P(0

,Y)

∆PEHLFRONT∆PEHL

REAR

∆PEHLCENT∆PEHL

LAT

Figure 3.13: Additional pressure amplitude definition for EHL contacts Φ =0.5, M = 400, L = 5

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3.3. EHL CONTACT PRESSURE 59

3.3.1 Indent crossing

−1 0 1

−0.5

0

0.5

1

X

−1 0 1

−0.5

0

0.5

1

X

−1 0 1

−0.5

0

0.5

1

X−1 0 1

−0.5

0

0.5

1

X

Psmooth

P∆P

Figure 3.14: Indent traversing an EHL contact M = 400, L = 5

Similar results as for dry contacts in section 3.2.1 have been obtained forEHL pure-rolling contacts. Figure 3.14 shows that the additional pressuredistribution remains almost constant when an indent traverses the contact.For pure-rolling conditions, the geometrical deformation in EHL contactshas been studied by several authors especially for harmonic waviness. Thetransition between the low and high pressure region is associated with animportant increase in the lubricant viscosity. The fluid becomes so viscousthat its geometry is “frozen”. The lubricant separating the two surfaces(indent, roughness. . . ) can no longer be deformed, it traverses the contactwithout any change in shape. For pure-rolling conditions, the lubricanttraverses the contact at the same velocity as the surfaces. That explainswhy the additional pressure is almost constant.

Table 3.4 gives the error of the additional pressure amplitude over therear shoulder when approximating the maximum additional pressure ampli-tude during indent crossing by the centered position value. For example, onecan see in table 3.4 that the error in the additional pressure amplitude overthe rear shoulder between a centered indent position and the real maximumfor an indent diameter Φ = 0.5 and an indent slope D/Φ = 0.5 is equal to2%.

Similar conclusions as in the dry contact study are obtained. However,

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60CHAPTER 3. DRY AND EHL PRESSURE IN INDENTED CONTACTS

D

Φ= 0.5

D

Φ= 1

Φ = 0.5 2% 4%Φ = 1 3% 6%

Table 3.4: Difference in the additional pressure amplitude overthe rear shoulder during indent crossing for EHL contacts =∣

∣∆PEHL

XD=0 − ∆PEHLmax(T )

∣/∆PEHL

max(T ) as a function of the indent diameter Φ

and the indent slope D/Φ for K = 4, M = 400, L = 5

two differences exist: calculations are still transient even if one considersonly the centered indent position and it is correct only for Newtonian pure-rolling conditions neglecting the outlet pressure spike.

3.3.2 Ellipticity

−1 −0.5 0 0.5 1

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

Y

Pκ=1.0

∆ Pκ=1.0

Pκ=0.1

∆ Pκ=0.1

Figure 3.15: Pressure distribution around an indent when varying the ellip-ticity in an EHL contact, Φ = 0.5, M = 400, L = 10

For EHL contacts, the ellipticity variation is made in accordance withthe dry contact study. The radius of curvature is changed according to theellipticity. As for dry contacts, this change appears in the equation 2.4, butalso in the equation 2.15.

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3.3. EHL CONTACT PRESSURE 61

D

Φ= 0.5

D

Φ= 1

M = 200 L = 6.3Φ = 0.50Φ = 1.00

< 1%< 1%

2%4%

M = 400 L = 10Φ = 0.50Φ = 1.00

1%< 1%

3%2%

Table 3.5: Difference in the additional pressure amplitude over the rearshoulder for two ellipticities κ = 1.0 and κ = 0.2 for EHL contacts =∣

∣∆PEHLκ=1.0 − ∆PEHL

κ=0.2

∣ /∆PEHLκ=1.0 as a function of the operating conditions M ,

L, the indent diameter Φ and the indent slope D/Φ for K = 4

Additional pressure distributions in indented contacts are hardly affectedby an ellipticity variation. Figure 3.15 gives an example of the pressure dis-tribution around an indent in a circular and an equivalent elliptical contact.Table 3.5 presents some examples of the relative difference between ellipticaland circular indented contacts as a function of the indent geometry and theoperating conditions. It can be noticed that the difference remains small andis of the same order of magnitude as the discretisation error. For example,for M = 400 and L = 10, Φ = 0.5 and D/Φ = 1.0, the difference betweenκ = 1.0 and κ = 0.2 is 3%. The ellipticity influence is thus neglected. Hookeand Venner [56] have presented results in accordance with this conclusion.The authors explain that when varying the ellipticity, the modification ofthe inlet flow leads to a negligible modification of the harmonic wavinessdeformation.

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62CHAPTER 3. DRY AND EHL PRESSURE IN INDENTED CONTACTS

3.3.3 Pure-rolling EHL pressure prediction

The important parameters of the indented lubricated contact study are pre-sented. The dry contact study and existing lubricated contact deformationtheory will be used and extended.

The amplitude reduction theory predicts the deformation of harmonicwaviness in EHL contacts. Some differences between the amplitude reduc-tion theory and the indented contact pressure study exist. The amplitudereduction theory studies harmonic waviness of small amplitudes. Typically,the initial waviness amplitude does not exceed 10% of the smooth centralfilm thickness, whereas, indent depth can exceed by a factor of ten or morethe film thickness. Under these conditions, the linear relation between de-formed and initial amplitude is no longer true. A second difference is thecomplexity of the 2-D indent geometry. Moreover, the harmonic wavinessdeformation is studied for a periodic signal. For indents, no “established”regime can be defined because indents are not periodic.

Depending on the operating conditions and the indent geometry, thepressure perturbation amplitude can vary substantially. This work presentsan analysis of the parameter influence and shows an approach to predictthese pressure perturbation amplitudes. Existing theory for roughness de-formation is extended to pure-rolling EHL indented contact pressure pertur-bations. First, a study of shallow indents is proposed, then sharper indentsare studied. Key parameters are presented permitting theoretical predic-tions.

Figure 3.16 is an example of the additional pressure amplitude over therear shoulder in an EHL contact as a function of the dry contact value.Different speeds have been represented (all the other parameters remain un-changed). When the speed decreases, M increases and L decreases. Twopoints are interesting: first, it shows the importance of the operating con-ditions because the additional pressure amplitude varies substantially withthe speed variation; second, for pure-rolling contacts the dry contact is anupper boundary. When decreasing the speed, the EHL additional pressureamplitude approaches the dry contact value.

Shallow indent

For small amplitudes, the harmonic waviness deformation is linear. It jus-tifies the use of the ratio between the deformed and the initial amplitudes.Hooke proposed a dimensionless parameter ∇, representing the ratio be-tween the waviness wavelength and the inlet contact length. This param-eter is proportional to λ/b ·

M/L, which corresponds to the parameterresulting from numerical studies conducted by Venner and Lubrecht.

The amplitude reduction curve presents two asymptotes: for small ∇, theroughness is associated with a small deformation (small pressure perturba-

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3.3. EHL CONTACT PRESSURE 63

0 1.0 2.0 3.0 4.00

1.0

2.0

3.0

4.0

∆ PDRY

/ PREF

∆ P

EH

L / P

RE

F

M = 100 L = 7.94M = 200 L = 6.30M = 400 L = 5.00M = 800 L = 3.97dry

Figure 3.16: Additional pressure amplitude over the rear shoulder in ehlcontacts compared to a dry contact when varying the mean speed, Φ = 0.5

tion), and large ∇ corresponds to a completely flattened geometry (pressureperturbation close to the dry contact). So, to account for these asymptotes,the curve fit presented is based on the following functions:

S∆PEHL

S∆PDRY∝ 1 − 1

1 + f(∇Φ)(3.7)

f(0) = 0

df(∇Φ)

d∇Φ> 0 for ∇ > 0

∇Φ ∝ Φ

M

L(3.8)

In the case of indents, the deformed amplitude is difficult to define.The deformed geometry is not harmonic, so the ratio between initial anddeformed amplitude is not clearly defined. That is an other reason why thepressure perturbation amplitudes (positive and negative) are used, insteadof the deformed amplitudes. For pure-rolling EHL contacts, the dry contactpressure is an upper limit.

The main idea is to assume that, similarly to small waviness deformation,the small indent additional pressure can be described as a function of theoperating conditions and the indent diameter (which corresponds to a pseudowavelength used to define ∇Φ). As mentioned before, indents can be very

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64CHAPTER 3. DRY AND EHL PRESSURE IN INDENTED CONTACTS

deep. In order to use the linearity of small indent pressure variations, theinitial slopes S∆PEHL of the curves ∆PEHL as a function of indent slopeD/Φ are used. For given operating conditions, if the indent slope tends tozero, it means that the indent depth will become much smaller than the filmthickness, given Φ constant.

S∆P =d∆P (D/Φ)

d(D/Φ)

D/Φ→0

(3.9)

The indent geometry and the associated transient phenomena are com-plex. The pressure perturbation amplitudes are not constant around theindent. That is why several sets of curves will be proposed to model theadditional pressure amplitudes.

Figures 3.17 and 3.18 recap results (initial slope) of numerous calcu-lations representing a relatively wide range of indent sizes and operatingconditions. Every point represents several hours of computing time. Asingle curve can be clearly observed. The pressure excursion slope ratioS∆PEHL/S∆PDRY increases with the dimensionless indent diameter pa-rameter ∇Φ

4. As mentioned above, the two asymptotes of the pressureperturbations also exist: on the one hand, contact conditions yielding smalldeformations and small pressure perturbations; on the other, the indent iscompletely flattened and the dry contact pressure amplitude is reached.

4The numerical value of S∆PDRY is known as a function of K from the dry contact

study

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3.3. EHL CONTACT PRESSURE 65

10−1

100

101

102

103

0

0.2

0.4

0.6

0.8

1

∇ Φ

S∆P

EH

LR

EA

R/S

∆PD

RY

Φ=0.25Φ=0.5Φ=1.0FIT

Figure 3.17: Relative initial slope S∆PEHLREAR/S∆PDRY as a function of

the dimensionless indent diameter parameter ∇Φ, for M = 100 − 1200,L = 3 − 19, Φ = 0.25 − 1.0, K = 3

10−1

100

101

102

103

0

0.2

0.4

0.6

0.8

1

∇ Φ

S∆P

EH

LF

RO

NT/S

∆PD

RY

Φ=0.25Φ=0.5Φ=1.0FIT

Figure 3.18: Relative initial slope S∆PEHLFRONT /S∆PDRY as a function of

the dimensionless indent diameter parameter ∇Φ, for M = 100 − 1200,L = 3 − 19, Φ = 0.25 − 1.0, K = 3

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66CHAPTER 3. DRY AND EHL PRESSURE IN INDENTED CONTACTS

10−1

100

101

102

103

0

0.2

0.4

0.6

0.8

1

∇ Φ

S∆P

EH

LC

EN

T/S

∆PD

RY

CE

NT

Φ=0.25Φ=0.5Φ=1.0FIT

Figure 3.19: Relative initial slope S∆PEHLCENT /S∆PDRY as a function of

the dimensionless indent diameter parameter ∇Φ, for M = 100 − 1200,L = 3 − 19, Φ = 0.25 − 1.0, K = 3

A similar analysis is possible for the negative pressure perturbation.However, as the behavior of the pressure in the indent hole is very non-linear, results are less satisfying. Nevertheless, several results are interest-ing. The initial slopes of numerous calculations have been extracted and arerepresented in figure 3.19.

Figure 3.20 enables one to predict the initial slope of the curves ∆PEHL

as a function of D/Φ. It represents in a single plot the three previousequations. This initial slope is a function of the indent diameter and theoperating conditions through the parameter ∇Φ. When ∇Φ increases, theratio S∆PEHL/S∆PDRY increases. When this ratio is close to 1, the ad-ditional pressure amplitude initial slope is close to the dry contact. Thethree curves are very similar and no clear difference can be extracted whenaccounting for the numerical accuracy.

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3.3. EHL CONTACT PRESSURE 67

10−1

100

101

102

103

0

0.2

0.4

0.6

0.8

1

∇ Φ

S∆P

EH

LF

IT/S

∆PD

RY

REARFRONTCENT

Figure 3.20: Relative initial slope S∆PEHL/S∆PDRY as a function of thedimensionless indent diameter parameter ∇Φ, for M = 100 − 1200, L =3 − 19, Φ = 0.25 − 1.0, K = 3

Moreover, the shoulder influence appears in figure 3.21, which comparesresults obtained previously (K = 3) with an indent with very flat shoulders(K = 8)5. The additional pressure amplitude initial slope is lower whenthe shoulders are flatter. However the two curves show similar trends andthis difference can also be partially associated with the slight hole geometryvariation when varying K.

The similarity of the curves in figure 3.21 does not mean that the pressureperturbations will be the same between K = 3 and K = 8. The curvesrepresent the ratio between S∆PEHL and S∆PDRY . Note that S∆PDRY

for K = 3 and K = 8 are very different. Hence, S∆PEHL between K = 3and K = 8 is also very different.

5The intermediate fit curves for K = 4 and K = 5 are also plotted

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68CHAPTER 3. DRY AND EHL PRESSURE IN INDENTED CONTACTS

10−1

100

101

102

103

0

0.2

0.4

0.6

0.8

1

∇ Φ

S

∆PE

HL

RE

AR

/S∆P

DR

Y

FIT K=3FIT K=4FIT K=5FIT K=8Φ=0.25Φ=0.5Φ=1.0

Figure 3.21: Relative slope S∆PEHLREAR/S∆PDRY as a function of the di-

mensionless indent diameter parameter ∇Φ and of the shoulder parameterK, for M = 100 − 1200, L = 3 − 19, Φ = 0.25 − 1.0

Sharp indent

The next step is to account for the non-linearity associated with deeper in-dents. The behavior of the additional pressure amplitude when increasingthe indent depth is complex and depends on the contact conditions. Ac-tually, the non-linear behavior is linked to the interaction between severalfactors. Complex piezoviscous effects are involved in the transition from thehigh pressure region over the indent shoulders to the medium pressure in theindent hole. The analysis is difficult because of important transient effects.Moreover, the fluid compressibility introduces an additional non-linearity.For the range of operating conditions and indent geometries used, figure3.22 represents the dispersion of the pressure perturbations over the frontshoulder ∆PEHL

FRONT . For readability, only a few operating conditions andindent geometries are reported. A similar dispersion can be observed for thepressure over the rear shoulder ∆PEHL

REAR in figure 3.23. Figure 3.24 repre-sents this dispersion concerning the negative additional pressure amplitudein the indent hole ∆PEHL

CENT . Only relatively sharp indents are represented.Concerning the negative additional pressure amplitude, a horizontal asymp-tote is relatively rapidly reached. It coincides with the pressure forming aplateau in the indent hole.

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3.3. EHL CONTACT PRESSURE 69

0 0.5 1 1.5 20

0.5

1

1.5

2

2.5

3

3.5

4

D/Φ

∆ P

EH

LF

RO

NT/P

RE

F

DRYM=100 L=7.94M=200 L=6.30M=1200 L=3.47M=100 L=15.87M=200 L=12.60M=800 L=11.91M=1200 L=10.40

Figure 3.22: Additional pressure amplitude over the front shoulder∆PEHL

FRONT /PREF for sharper indents K = 4. Blue curves for Φ = 0.25,red curves for Φ = 0.5 and green curves for Φ = 1

0 0.5 1 1.5 20

0.5

1

1.5

2

2.5

3

3.5

4

D/Φ

∆ P

EH

LR

EA

R/P

RE

F

DRYM=100 L=7.94M=200 L=6.30M=1200 L=3.47M=100 L=15.87M=200 L=12.60M=800 L=11.91M=1200 L=10.40

Figure 3.23: Additional pressure amplitude over the rear shoulder∆PEHL

REAR/PREF for sharper indents K = 4. Blue curves for Φ = 0.25,red curves for Φ = 0.5 and green curves for Φ = 1

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70CHAPTER 3. DRY AND EHL PRESSURE IN INDENTED CONTACTS

0 0.5 1 1.5 20

1

2

3

4

5

D/Φ

∆ P

EH

LC

EN

T/P

RE

F

M=100 L=7.94M=200 L=6.30M=1200 L=3.47M=100 L=15.87M=200 L=12.60M=800 L=11.91M=1200 L=10.40

Figure 3.24: Additional pressure amplitude in the indent hole∆PEHL

CENT /PREF for sharper indents K = 4. Blue curves for Φ = 0.25,red curves for Φ = 0.5 and green curves for Φ = 1

The additional pressure is divided by the predicted initial slope fromcurve fits presented previously in the shallow indent study S∆PEHL

FIT . Anasymptote appears logically for shallow indents. Figure 3.25 enables one topredict the pressure perturbations around sharp indents as a function of theindent geometry and the operating conditions.

Some efforts have been made to quantify the non linear effects whenincreasing the indent slope as a function of the operating conditions. How-ever, the accuracy of the initial slope prediction and of the calculationsthemselves have not permitted a noticeable improvement in the final dis-persion in figure 3.26 and 3.27. Moreover, these limited improvements leadto complex parameters without any physical interpretation. Figures 3.26and 3.27 represent the correlation between full numerical results and fit-ted model predictions. The obtained correlation is considered satisfactoryregarding the problem complexity.

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3.3. EHL CONTACT PRESSURE 71

0 0.5 1 1.5 20

0.5

1

1.5

2

2.5

D/Φ

∆ P

EH

Lm

odel

/ S

∆ P

EH

LF

IT

model rear K=3−8model front K=3model front K=4model front K=5model front K=8

Figure 3.25: ∆PEHLFRONT /S∆PEHL

FRONT FIT and ∆PEHLREAR/S∆PEHL

REAR FIT fordeeper indents. M = 100 − 1200, L = 3 − 19, Φ = 0.25 − 1 and K = 3 − 8

0 1 2 3 40

1

2

3

4

∆ PEHLFRONT model

/ PREF

∆ P

EH

LF

RO

NT

num

eric

al /

PR

EF

Figure 3.26: Full numerical prediction versus model predictions of the ad-ditional pressure amplitude over the front shoulder ∆PEHL

FRONT /PREF forM = 100 − 1200, L = 3 − 19, Φ = 0.25 − 1, D/Φ = 0 − 2 and K = 3 − 8

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72CHAPTER 3. DRY AND EHL PRESSURE IN INDENTED CONTACTS

0 1 2 3 4 50

1

2

3

4

5

∆ PEHLREAR model

/ PREF

∆ P

EH

LR

EA

R n

umer

ical

/ P

RE

F

Figure 3.27: Full numerical prediction versus model predictions of theadditional pressure amplitude over the rear shoulder ∆PEHL

REAR/PREF forM = 100 − 1200, L = 3 − 19, Φ = 0.25 − 1, D/Φ = 0 − 2 and K = 3 − 8

3.3.4 Sliding influence

Even if tapered roller bearings operate very close to pure-rolling, sliding ex-ists (almost negligible). Moreover, other bearing technologies lead to slidingwhich can reach about 5%. The Newtonian fluid assumption can be notrealistic in these cases. However, Newtonian calculations are a asymptoticcase for high shear limit fluids. This section propose a limited study of thesliding influence. Only some major trends will be presented. Several authorspresent sliding as an important parameter Kaneta et al. [70], Ai and Cheng[1], Ai and Lee [2], Ai and Nixon [3, 4], Ville and Nelias [121], Nelias andVille [91], Diab et al. [31]. The examples below show how a small amountof sliding of 4% can double the additional pressure amplitude.

Figure 3.28 represents an example of the pressure distribution for a cen-tered indent for different slide to roll ratios. The slide to roll ratio is definedas the speed difference between the dented and the smooth surface dividedby the velocity sum:

uD − uS

uD + uS(3.10)

One can see that the pressure distribution is noticeably affected by sliding.Sliding has an obvious detrimental effect because of larger pressure pertur-

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3.3. EHL CONTACT PRESSURE 73

bations. In this example, the pressure amplitude is almost doubled betweenpure-rolling and 4% sliding conditions.

−1.5 −1 −0.5 0 0.5 1 1.50

0.2

0.4

0.6

0.8

1

1.2

X

P

+4%+2%0%

Figure 3.28: Pressure distribution for a centered indent for different slideto roll ratios 0%, +2%, +4%. Indent on the fast surface Φ = 0.4, M = 500,L = 8

When comparing different slide to roll ratios, the mean speed remainsthe same and the indent is on the slow or the fast surface. The induceddent phenomenon can not be observed clearly with such small slide to rollratios. However, the same physical phenomenon occurs: the lubricant be-comes extremely viscous when pressure increases and the lubricant travelsat the mean speed. When the lubricant geometry is shifted from the sur-face geometry, the edge of the surface indent is deformed by the lubricantgeometry. It leads to a larger pressure perturbation.

However, even if this numerical observation was confirmed by experi-mental film thickness measurements, the impact of sliding on life seems tobe more complex. Several authors present experimental results where thefailure initiation spots are not in accordance with the largest calculatedpressure and stress locations. Sliding seems to be an important parameter;however, the numerical model used in this work fails to correctly explain alldetails of the experimental observations. The role of the residual stresses,the friction forces and the crack propagation seem to be required to explainthe life results in sliding conditions (whereas these aspects are neglected inthis work). Kaneta et al. [70] studied the crack propagation directions. Theorientation of the friction forces appears to be determinant. Moreover, thehydraulic pressure influence (when oil is trapped in the crack) can increasethe crack propagation speed.

Sliding studies are more complex than pure-rolling conditions. For ex-ample, the pressure maximum above indent shoulders can no longer be ap-

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74CHAPTER 3. DRY AND EHL PRESSURE IN INDENTED CONTACTS

proximated by the centered indent value. Indeed, this maximum appearsat a different indent location. Figure 3.29 represents the pressure distribu-tion at different indent locations. The additional pressure amplitudes are nolonger constant during the indent crossing, the dash-dotted and the dottedcurves represent the additional pressure over the front and the rear shoulderduring the crossing. Jacq [65] presents an experimental work using severalartificial indent sizes and operating conditions under pure-rolling or smallslide to roll ratios. The author showed that the relative severity ranks of thedifferent indents remain unchanged when introducing a small slide to rollratio. As the model developed here will be adjusted to experimental data,one can expect that the pure-rolling calculations will be able to describethe relative severity of indents observed on bearing surfaces. Finally, in thiswork, the indent geometry is axysymmetric. It is already a simplificationof the real indent geometry. Sliding conditions lead to much more complexnatural indent geometries and pressure distributions.

−1 0 1−1

0

1

X

−1 0 1−1

0

1

X

−1 0 1−1

0

1

X−1 0 1

−1

0

1

X

P∆ P

Figure 3.29: Indent traversing a +4% rolling sliding EHL contact. Indenton the fast surface Φ = 0.4, M = 500, L = 8

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3.4. CONCLUSION 75

3.4 Conclusion

Several results have been described concerning the pressure distribution inindented contacts. First, the additional pressure distribution is almost con-stant when an indent traverses a contact. This is both true for dry andpure-rolling EHL contacts. Therefore, most of the pressure results are pre-sented for a centered indent. This observation will be used in the nextchapter for the stress calculation. Second, some dimensionless elliptical pa-rameters have been presented. These parameters enable one to generalizethe pressure results obtained in circular contacts to wide elliptical contactsfor sufficiently small indents. One dimension (especially time consuming) ofthe parametric study is therefore avoided. Moreover, an analytical modelof the dry contact pressure perturbation was presented. Finally, curves pre-dicting pure-rolling EHL additional pressure amplitude as a function of theindent geometry and the operating conditions were proposed. The detri-mental effect of the (small) slide to roll ratio has been briefly presented.

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76CHAPTER 3. DRY AND EHL PRESSURE IN INDENTED CONTACTS

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Chapter 4

Indented contact stress and

life

4.1 Introduction

The goal of this chapter is to propose a life model of indented bearings. Thestudy of the indented contact pressure distribution in the previous chapterallowed to outline and to model some important phenomena. The step be-tween pressure distribution and component fatigue life is not trivial. Therolling contact fatigue behavior is complex. Test campaigns are expensive(and time consuming) and lead to relatively large experimental dispersions.The numerous existing models are rarely universal and usually require ex-perimental adjustments. It is not that easy to prove statistically that amodel is better suited than another. A very brief presentation of life modelsis given; a discussion about the stress criteria is proposed. Some convenientapplications of the conclusions of chapter 3 are proposed to facilitate thestress and life calculation. A compromise between computing time, memoryrequirement and numerical accuracy is presented. Moreover, stress and lifeprediction results are presented and a life model is proposed.

The life prediction model developed has to be correlated and adjusted toexperimental data. An experimental adjustment procedure is briefly men-tioned. Some experimental results concerning “natural” indentation statis-tics are proposed. An example is detailed.

77

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78 CHAPTER 4. INDENTED CONTACT STRESS AND LIFE

4.2 Contact stress and life

4.2.1 Stress and life criteria

The material (steel) is far from simple. At a sufficiently small scale, it isvery heterogeneous. Many defects are present. The defects vary substan-tially in terms of nature, size, depth and finally severity. Fatigue failure isassociated with crack initiation and finally spalling. Defects play the role ofstress raisers and cracks are initiated in their vicinity. These weak pointsare randomly distributed throughout the material. Depending on the load-ing cycle, these defects are activated or not. Dudragne and Girodin [34],Piot et al. [97] present a model which is based on material defect statisticsand crack propagation calculations. However, the defect “severity” is noteasy to evaluate. The microscopical behavior modeling often requires theidentification of material parameters.

Some macroscopic fatigue criteria assume that the time between crackinitiation and its propagation is sufficiently small to be neglected. Therandom distribution of material defects can be modeled using a statisticalWeibull approach. The Weibull approach was developed for rolling contactfatigue (RCF) by Lundberg and Palmgren [87]. The authors experimentallylinked the failure probability F to the τxz stress amplitude under Hertzianloading, to the stressed volume v and to the maximum stress depth z, seeequation 4.1. Four parameters are adjusted to fit the experimental data:the stress exponent c, the Weibull slope e, the depth exponent h and theconstant cst.

F = 1 − cst Le τ cxz v

zh(4.1)

Ioannides and Harris [64] propose a criterion which is the extension ofLundberg and Palmgren’s work. The stress distribution is integrated overthe volume instead of considering only its maximum, see equation 4.2. Thedimensionless risk integral is defined in equation 4.4. Since then, the studyof various surface patterns or defects was possible. The notion of infinitelife was also introduced. Lubrecht et al. [82] demonstrated that the depthterm zh had no physical meaning and lead to a singular behavior close tothe surface.

F = 1 − cst Le

∫∫∫

Ω

max (τ − τ∞ ; 0)c

zhdxdy dz (4.2)

I =

∫∫∫

Ωmax (τ − τ∞ ; 0)c dxdy dz = pc

Hb3Ic (4.3)

Ic =

∫∫∫

Ωmax (T − T∞ ; 0)c dX dY dZ (4.4)

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4.2. CONTACT STRESS AND LIFE 79

Several choices are possible concerning the stress criterion. Tresca andVon Mises are commonly used in RCF studies. Actually, these criteria aremore plasticity than fatigue criteria. Most recent studies are based on moresophisticated multiaxial fatigue criteria. Multiaxial criteria, for exampleDang Van [29] or Papadopoulos [95], are based on the stress variation am-plitude in every material point. These multiaxial criteria consider that everymaterial point is submitted to a stress cycle. At a microstructural scale, thematerial accommodates this stress cycle by changing its microstructure. Thevariation around this stabilized stress state is studied for fatigue predictions.A coupling with the hydrostatic stress is then made 1.

X

Z

−1 0 1−1

−0.5

0

X

Z

−1 0 1−1

−0.5

0

X

Z

−1 0 1−1

−0.5

0

X

Z

−1 0 1−1

−0.5

0

X

Z

−1 0 1−1

−0.5

0

X

Z

−1 0 1−1

−0.5

0

0 0.5 1 1.5a b

c d

e f

Figure 4.1: Stress distribution σV M/σV MS max at different indent locations a-

d, and stress amplitude history e-f in the vertical plane along the rollingdirection for a circular dry contact (e smooth, f dented), Φ = 0.5, K = 3

1Without entering in too many details, the residual stresses (manufacturing, runningin. . . ) or constant stress like hoop stress play no role in the variation around the stabilizedstress state. Actually, they appear only in the hydrostatic pressure coupling.

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80 CHAPTER 4. INDENTED CONTACT STRESS AND LIFE

Figure 4.1 shows the stress amplitude history around an indent. Thisfigure represents the stress distribution at different indent locations (plots4.1 a-d) in a vertical plane along the rolling direction X. For every contactlocation, the stress distribution in the volume is calculated. One can seethat the pressure perturbation on the surface due to an indent induces ahigh stress zone close to the surface. For a simple criterion like Von Mises,the stress amplitude history is defined as the maximum stress endured byeach point during the contact crossing. For a multiaxial criterion, a moresophisticated treatment is used. However, it is shown further on that rela-tively similar results are obtained. If a smooth contact is used, horizontalisovalues are obtained, see plot 4.1 e. For an indented contact, one ob-tains a more complex distribution, see plot 4.1 f. To be able to comparethe stress distribution between different criteria, every stress value has beennormalized by the smooth contact maximum stress amplitude.

Several remarks concerning RCF predictions can be made. First, theDang Van or Papadopoulos multiaxial criterion evaluation is much morecomplex and requires to calculate and to store the stress of every mate-rial point at every time step. Second, RCF tests are submitted to suchlarge confidence intervals that it is very difficult to prove the benefits in lifeprediction accuracy from one criterion to an other, unless their predictionsdiffer significantly. However, if the beneficial coupling effect of high com-pressive hydrostatic stress is generally accepted, Desimone et al. [30] showthat under certain condition high compressive hydrostatic stresses have anadverse effect on fatigue life. Finally, the RCF life predictions are not madeas “absolute” life, but with respect to a material constant which is experi-mentally adjusted. If different criteria give “homothetical” stress histories,the material constant will more or less integrate the different proportionalitycoefficients. It is especially true for the Tresca, Von Mises, Dang Van andPapadopoulos criteria used in Hertzian smooth rolling contacts.

Figure 4.2 represents the stress amplitude history for Papadopoulos’ mul-tiaxial criterion around an indent in a dry contact in a vertical plane alongthe rolling direction axis X. The indent is centered in X = Y = 0. One cansee that far from the indent, the maximum value returns logically to 1. In-deed, the stress perturbations due to the indent are sufficiently far to vanishand the stress distribution becomes that of a smooth contact. The stressamplitude history for the Papadopoulos multiaxial criterion looks similar tothe stress distribution obtained with a simple Von Mises criterion in figure4.1 f. A more detailed comparison is made in the next section.

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4.2. CONTACT STRESS AND LIFE 81

X

Z

−1.5 −1 −0.5 0 0.5 1 1.5−1

−0.8

−0.6

−0.4

−0.2

00 0.5 1 1.5

Figure 4.2: Indented contact stress amplitude history σPA/σPAS max for Pa-

padopoulos’ multiaxial criterion for a circular dry contact, Φ = 0.5, K = 3

4.2.2 Pseudo-transient calculation

X

Z

−1.5 −1 −0.5 0 0.5 1 1.5−1

−0.8

−0.6

−0.4

−0.2

00 0.5 1 1.5

Figure 4.3: Indented contact stress amplitude history σV MPT /σV M

S max in thevertical plane along the axis X for a pseudo-transient calculation for a cir-cular dry contact, Φ = 0.5, K = 3

Pure elastic deformation is assumed. The associated stress distributionis an elastic stress field. This means that the stress tensor can be linearlydivided into two tensors using the superposition principle. A considerableshortcut can be made if this decomposition is chosen carefully.

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82 CHAPTER 4. INDENTED CONTACT STRESS AND LIFE

σtot = σS + σD (4.5)

Chapter 3 presented two important results for the stress calculation.First, the additional pressure distribution is constant during the indentcrossing. Second, the ellipticity hardly modifies this additional pressuredistribution for sufficiently small indents. This leads to two interesting re-sults:

First, the complete stress history around an indent can be emulatedusing only the centered indent stress distribution and the smooth stressdistribution. This will be referred to as a “pseudo-transient” calculation.Instead of calculating and storing the pressure values at every time stepand also the stress distribution, only two time steps are needed: the one ofsmooth contact and the centered indent. It represents an important gain incomputing time and memory2.

Second, the smooth stress distribution can be replaced by the smoothstress distribution of any ellipticity3. It leads to the same conclusions as forthe pressure. The elliptical analysis, an especially time consuming parameterbecause of the number of unknowns, is thus avoided.

Of course, the pseudo-transient calculation has a drawback in terms ofresult accuracy. Moreover, some special care is needed when the two stressdistributions are assembled emulating every time step. For example, theadditional stress distribution σD has absolutely no meaning when the indentis out of the contact.

Figure 4.3 represents the stress history around an indent in a verticalplane along the rolling direction X for a simple Von Mises pseudo-transientapproximation. The corresponding complete Von Mises calculation is rep-resented in figure 4.1 f and the Papadopoulos multiaxial calculation in fig-ure 4.2. The stress distributions are relatively similar. The complete VonMises calculation and pseudo-transient approximation are almost identical.A slight difference is found in the stress distribution under the indent hole. Itcan undoubtedly be neglected because the stress is especially small in thisvery region. Concerning the comparison between the multiaxial criterionand the pseudo-transient approximation, the difference is more significant.The stressed volume seems more important for the Papadopoulos criterion.However, the maximum stress value is similar. The maximum stress has animportant weight in the stress risk because of the power c in the equation4.2.

2Typical calculations use a 256 × 256 grid on the surface, an equivalent of about 24grids in depth for each stress tensor component and about 256 time steps. These twostress distributions are artificially shifted to simulate the indent movement. It representsabout 2.5 billion numbers (20 Go of data if all the values are needed as in the Dang Vanor Papadopoulos criteria).

3The infinitely elongated contact stress distribution is detailed in appendix C.

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4.2. CONTACT STRESS AND LIFE 83

D/Φ = 1 D/Φ = 2

Φ = 0.25K = 3K = 8

5%3%

6%4%

Φ = 0.5K = 3K = 8

8%7%

7%7%

Table 4.1: Difference between different stress criteria(

σPAD max/σPA

S max − σV MD max/σV M

S max

)

/(

σV MD max/σV M

S max

)

as a function ofthe indent diameter Φ, the indent slope D/Φ and the decay coefficient Kfor dry contacts

Table 4.1 represents the comparison between two stress criteria (Pa-padopoulos’ multiaxial and Von Mises) of the maximum indented contactstress amplitude normalized by the maximum smooth contact stress ampli-tude as a function of the indent geometry. The Von Mises complete calcu-lation is used as a reference. The difference between a Von Mises completecalculation and a Von Mises pseudo-transient approximation is less than1%, thus it is not reported. The Papadopoulos and the Von Mises casesgive similar variations. For this study and the range of parameters used,the multiaxial Papadopoulos criterion is systematically more severe. Forexample, the difference between the Papadopoulos multiaxial criterion andthe Von Mises criterion for an indent diameter Φ = 0.25, an indent slopeD/Φ = 1 and a shoulder parameter K = 3 is 5%.

The difference is limited and only much better experimental campaignscan reduce experimental confidence intervals and justify a more sophisticatedstress criterion. The goal is only to show that the different criteria lead tosimilar results4. The computational effort of a multiaxial criterion was notjustified regarding the assumptions and above remarks. Thus a simple VonMises criterion with a pseudo-transient approximation is used.

4.2.3 Stress and life prediction

The additional pressure amplitude study was justified for several reasons.First, the maximum elastic stress is linearly proportional with the additionalpressure amplitude. Figure 4.4 represents the maximum stress σD max/σS max

as a function of the additional pressure amplitude ∆P/PREF for dry con-tacts. This figure confirms that the maximum stress varies linearly as afunction of the additional pressure amplitude. The results from chapter 3concerning the additional pressures are thus relevant to the contact sever-

4Note that the residual stress influence is not investigated here. Lubrecht et al. [85]show some results using the indentation residual stress. They conclude that for pure-rolling conditions and the obtained indent geometries, the indentation residual stress isbeneficial because of high compressive stress under the indent shoulders.

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84 CHAPTER 4. INDENTED CONTACT STRESS AND LIFE

ity. Second, the additional pressure study enables one to have a more directunderstanding of the different phenomena. An important part of chapter 3is based on the shallow indent pressure perturbation study. The equivalentstudy in terms of stress would be difficult.

0 1 2 3 4 5 61

1.2

1.4

1.6

1.8

2

2.2

∆ P / PREF

TD

max

/ T S

max

Φ=0.25Φ=0.50Φ=1.00FIT

Figure 4.4: Maximum stress TD max/TS max as a function of the additionalpressure amplitude ∆P/PREF for dry contacts, Φ = 0.25 − 1.0, D/Φ =0.25 − 2 and K = 3 − 8

The reader has probably noticed that most results are presented usingdimensionless parameters. When the dimensional maximum stress is cal-culated from its dimensionless value, a question can arise: How can thecalculated dimensional stress exceed the material plasticity limit? Figure4.4 shows that some of the calculated dimensionless stresses reach twice themaximum Hertzian stress for the range of parameters used in this study.If this particular indent was measured from a contact run under a 2.5 GPaHertzian pressure, it would mean that it could induce a shear stress closeto 1.8 GPa. A common plasticity limit for bearing steel is 1.5 GPa. Thisparticular indent geometry can not exist in this contact. It would have beendeformed during the running in process until the induced stress is lowerthan the local plasticity limit (including isotropic and kinematic harden-ing). However, if this indent geometry is effectively measured after runningin, it means that it does not induce any plastic deformation. The ma-terial has been locally hardened and the residual stresses are opposed tothe stress distribution associated with the indent deformation. The globalstress distribution does not lead to plastic deformation. Moreover, the stressamplitude endured by each material point during the loading cycle can beapproximated by the elastic stress amplitude.

Figure 4.5 represents the dimensionless risk integral ID/IS as function

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4.2. CONTACT STRESS AND LIFE 85

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 21

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3

D/Φ

I D /

I S

Φ=0.25 K=3 T∞ / T

S max=0.70

Φ=0.50 K=3 T∞ / TS max

=0.70

Φ=0.50 K=8 T∞ / TS max

=0.70

Φ=1.00 K=3 T∞ / TS max

=0.70

Φ=0.50 K=3 T∞ / TS max

=0.35

Figure 4.5: Dimensionless risk integral ID/IS as a function of the indent ge-ometry Φ, D/Φ and K and of the dimensionless infinite life limit T∞/TS max

for dry contacts

of the indent geometry K, D/Φ, Φ and the dimensionless infinite life limitT∞. The curves presented in figure 4.5 show that all these parameters havea significant impact on the dimensionless risk integral. The indent geometryinfluences both the stress amplitude and the stressed volume. For example,the first, second and fourth curves show how the stressed volume (increasingwith Φ) influences the risk integral. The second and third curves comparetwo different shoulder shapes. Finally, the second and fifth curves comparetwo T∞/TS max ratios. The dimensionless infinite life limit influences therelative weight between the high stressed zone under the indent shouldersand the Hertzian smooth contact stressed zone. The highly stressed zoneinfluence increases with this dimensionless infinite life limit (for exampledecreasing the Hertzian pressure).

The detrimental impact of an indent increases (relatively) for low Hertzianpressures. First, the indent slope and diameter are inversely proportionalto the contact half width. Second, the dimensionless infinite life limit in-creases when decreasing the Hertzian pressure. These two points increasethe dimensionless risk integral ratio.

The stress distribution can be schematically divided into two contribu-tions: an annular highly stressed zone close to the surface due to the indentpressure perturbations and a Hertzian smooth contact stress distributionzone. The following equation is proposed to approximate the corresponding

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86 CHAPTER 4. INDENTED CONTACT STRESS AND LIFE

risk integral:

(

ID

IS

)c

= qK(K) · qT∞

(

T∞

TS max

)

· Φ3 ·

TD maxTS max

− T∞TS max

1 − T∞

TS max

c

+ 1 (4.6)

The functions qK(K) and qT∞(T∞/TS max) in equation 4.6 are adjusted

to fit the numerical results. They account respectively for two points: first,the highly stressed volume depends on K (because the additional pressurehalf width is a function of K); second, the stress integration domain is afunction of T∞/TS max and an adjustment is required. Figures 4.6 a, b andc represent respectively the function qK(K), the function qT∞

(T∞/TS max)and IS/IS T∞=0 as a function of T∞/TS max. The figures and the equation4.6 allow to predict the dimensionless risk integral for dry contacts. Themaximum stress TD max is known from the figure 4.4 and finally from theadditional pressure amplitude predictions of chapter 3. The correlation be-tween full numerical calculations and the simplified model is good, see figure4.7.

0 0.5 10

0.02

0.04

0.06

T∞ / TS max

q T∞

0 5 10

0.7

0.8

0.9

1

K

q K

0 0.5 10

0.5

1

T∞ / TS max

I S /

I S T

∞=

0

a b c

Figure 4.6: Adjustment functions qK(K) (plot a) and qT∞(T∞/TS max) (plot

b), dimensionless risk integral IS/IS T∞=0 as a function of the dimensionlessinfinite life limit T∞/TS max for dry contacts (plot c)

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4.3. EXPERIMENTAL INDENTATION AND LIFE CALCULATION 87

1 1.5 2 2.5 31

1.5

2

2.5

3

( ID

/ IS )

model

( I D

/ I S

)nu

mer

ical

Figure 4.7: Full numerical dimensionless risk integral as a function of themodel prediction for dry contacts, c = 6.3 − 11.3, T∞/TS max = 0.3 − 0.7,Φ = 0.25 − 1.0, K = 3 − 8

4.3 Experimental indentation and life calculation

4.3.1 Natural indentation and model adjustment

Surface indentation from lubricant pollution is a very complex process.Chapters 1 and 2 presented some of the difficulties encountered for theindentation prediction. For example, particle nature and geometry varysubstantially; operating conditions (speed, slide to roll ratio. . . ) and lubri-cant supply (particle mixing. . . ) are decisive but very difficult to evaluate.The aim of this work is not to try to predict bearing life from oil clean-liness. Our analysis starts with the indented surface geometry measuredusing interferometric microscopy.

Two possibilities exists: first, the indentations are formed in a real device(gear box. . . ); second, reference pollution test conditions are used to indentthe surfaces. Both methods enable one to sidestep the indentation predictionissue. The first method yields representative results of pollution in a realapplication. The second is well suited to compare different bearings on acommon basis. For example, the influence of the heat treatment in a pollutedenvironment can be studied this way.

The standard pollution test consists of running several bearings in acontrolled polluted environment. The size class, the particle material andthe particle concentration are chosen in order to reproduce indentationsobserved for example in an automotive gear box. The damage should beappropriate to correctly evaluate the indented bearing lives. After the in-dentation procedure, the bearings are cleaned and mapped. Many spots on

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88 CHAPTER 4. INDENTED CONTACT STRESS AND LIFE

the surface are measured. Pit statistics (geometry, density, location) areobtained from the numerical treatment of the surface mapping. Then, thebearings are life tested and compared to clean bearing lives using a standardlife test procedure. A correlation between measured surface indentation andcomponent lives is made. The numerical constants are adjusted to theseexperimental results.

0 0.5 10

0.01

0.02

0.03

0.04

0.05

probability

d/φ

0 0.5 10

50

100

150

200

250

300

probability

φ (µ

m)

Figure 4.8: Indent slope d/φ (left) and indent diameter φ (right) cumulativedistribution from a standard pollution test

A common ratio between clean and standard pollution life tests5 with90 − 125 µm ductile particles is a factor 10. Figure 4.8 shows indent pa-rameters of indented bearing topography results. It represents on the left,the slope and on the right, the diameter sorted in descending order. Forexample, one can see that 50% of the measured pits have a slope exceeding6

0.018 and that 50% of the measured pits have a diameter exceeding 85 µm.These median values should be taken with precaution. They depend sensi-bly on the surface mapping interpretation. Distinguishing roughness fromflat indents is not always trivial7.

Figure 4.9 (left) represents the correlation between the indent diameterand the indent slope. For each class of indent slopes, the maximum, themedian and the minimum indent diameters are plotted. One can clearly seethat the sharpest indents have the smallest diameter. For the 0.02 indentslope class, indent diameters are distributed between 50 and 180µm, themedian value is around 80µm. This example agrees with Ville [119] table

5Depends obviously on many parameters as particle concentration, operating condi-tions. . .

6corresponding to 2

7Several criteria are used: depth, circularity. . .

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4.3. EXPERIMENTAL INDENTATION AND LIFE CALCULATION 89

2.1 where the sharpest indents have the smallest diameters and correspondto the smallest particles.

However, these results are obtained in an experimental device where par-ticles pass the contact only once. Under Ville’s conditions, the indent slopedispersion was small. In a rolling element bearing, a particle is trappedin the bearing and it is laminated several times. Therefore, even for theunique particle size class used here, the indent slopes vary between zero andthe maximum slope. The smallest sharp indents correspond certainly to thefirst particle squashing and large shallow indents to the next overrollings.Figure 4.9 (right) represents the probability of every indent slope class usedfor the left plot. The cumulative distribution and density probability func-tions are plotted. For example, the 0.02 indent slope class represents 35% ofthe entire indent population and 70% of the entire indent population belongsto this class or a shallower indent slope class.

0 0.02 0.04 0.060

50

100

150

200

250

300

d/φ

φ (µ

m)

medianminmax

0 0.02 0.04 0.060

0.2

0.4

0.6

0.8

1

d/φ

prob

abili

ty

densitycumulative

Figure 4.9: Indent diameter as a function of the indent slope (left) and prob-ability of every indent slope class (right) from a topography measurementof a standard pollution test

The indent distribution of inner and outer ring are very close. The innerring indents seem slightly larger and deeper, perhaps due to a slightly higherHertzian pressure. Rolling bearings have specific kinematics: rotating orfixed inner and outer rings, and cage and roller relative motion. The numberof indents (or the indent density) can be estimated using these kinematics:overrolling contact frequency and loaded surface. This study is not detailedhere, as the results are not completely satisfying. The inner ring densityprediction is too small compared to measurements.

An example of common contact conditions is detailed below. It is a sim-plification of contact conditions in a bearing. For clarity, a unique Hertzian

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90 CHAPTER 4. INDENTED CONTACT STRESS AND LIFE

pressure amplitude and approximated operating conditions are used. How-ever, these values are representative of common contact conditions in ta-pered roller bearings. The mean bearing cone diameter is set to 5 cm andthe mean roller diameter to 7.8 mm. The bearing rotates at 2000 revs/mn.

Rx κ=∞ ≈ 3.9e-3mE′ = 226e9Pab ≈ 152e-6mpH ≈ 2.2e9PaRx κ=1 ≈ 5e-3m

T∞

TS max≈ 0.7

α = 20e-9Pa−1

η0 = 40e-3Pa.sus ≈ 5.9 m/sM ≈ 350L ≈ 17α ≈ 44

λ ≈ 4.5e-3

The dimensionless median indent slope is 0.6. The dimensionless medianindent diameter is 0.56. One can now associate a dimensionless risk integralto the indent geometry and the contact conditions. For a dry contact as-sumption, figure 4.10 represents the dimensionless risk integral distributionas a function of the dimensionless indent slope and the dimensionless indentdiameter. The color of each point represents the risk integral value8. Redpoints indicate the most severe indents.

The second step is to recalculate the risk integrals using EHL conditions.The operating conditions and the indent geometry lead to intermediate ∇Φ.It means that the lubricant plays an important role and modifies significantlythe pressure and stress distributions. Figure 4.11 is similar to figure 4.10using dimensionless risk integrals for EHL contacts. The most dangerousindents are shifted towards larger diameters. The explanation is straightfor-ward using the results from chapter 3. A larger indent diameter increasesthe ∇Φ value and the pressure perturbations tend to the dry contact val-ues. However, the color scale is deceptive. As explained, for the range ofparameters used in this work, the pure-rolling EHL contacts are less severethan dry contacts. If the same color scale were used for figures 4.10 and4.11, figure 4.11 would show only blue points. The lubricated contact riskintegrals certainly underestimate the contact risk in comparison with drycontact results9.

8Several points which are very close in terms of slope and diameter can have differentcolors because of different shoulder shapes K.

9The indentation data is based on measured surface topographies and yields a signifi-cant experimental life reduction under well lubricated conditions. So, both the operating

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4.3. EXPERIMENTAL INDENTATION AND LIFE CALCULATION 91

0 0.5 1 1.5 20

0.5

1

1.5

2

D/Φ

Φ

Figure 4.10: Dimensionless risk integral ID/IS as a function of the indentgeometry for a dry contact applied to a topography measurement of a stan-dard pollution indentation test (90 − 125 µm ductile particles)

0 0.5 1 1.5 20

0.5

1

1.5

2

D/Φ

Φ

Figure 4.11: Dimensionless risk integral ID/IS as a function of the indentgeometry for an EHL contact applied to a topography measurement of astandard pollution indentation test (90 − 125 µm ductile particles)

conditions and the indent statistics are realistic. Tapered roller bearings run very close topure-rolling; measured indents are relatively axisymmetrical (an other indirect confirma-tion of pure-rolling). A non Newtonian rheology would smooth pressure excursions andunderestimate EHL predictions even more. Experimental tests are conducted ensuring animportant lubricant supply; thus contacts are not starved. . .

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92 CHAPTER 4. INDENTED CONTACT STRESS AND LIFE

The adjustment to experimental data uses the same type of analysis. Themore complex bearing operating conditions and bearing kinematics (rotat-ing ring or not, loading zone. . . ) is accounted for. A Weibull probabilisticapproach is used to calculate the life reduction due to the surface indenta-tion.

4.3.2 Life calculation

The life calculation with a Weibull approach is relatively simple since the riskintegrals are known as a function of the indent geometry and the operatingconditions. Considering a non rotating ring and a constant loading, eachpoint of the ring is submitted to a constant stress cycle. The ring can bemeshed using elementary surfaces Si. Every surface Si is associated with arisk integral Ii. The Ii distribution is made accordingly to the indentationstatistics and the operating conditions over surfaces Si. The life Lϑi isdefined as the life which corresponds to a failure probability ϑ before Lϑi.

The failure probability Fi and the Lϑi life for surfaces Si assuming aconstant Weibull slope e reads :

Fi = 1 − eln(1 − ϑ)

(

L

Lϑi

)e

(4.7)

Leϑi ∝

1

Ii∝ 1

pcHi b

3i Ic

i

(4.8)

The surface S fails when the first elementary surface Si fails, it corre-sponds to a product of survival probabilities. The entire surface Lϑ lifereads :

Leϑ =

1∑ 1

Leϑi

(4.9)

Finally, it is possible to write the ratio between the dented surface lifeand the smooth surface life :

(

LϑS

)e

=

pcHi b

3i Ic

Si∑

pcHi b

3i Ic

i

(4.10)

An adjustment to experimental life tests can be made using the param-eter c. This parameter appears explicitly in the terms pc

Hi and implicitly inthe dimensionless risk integrals Ii.

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4.4. CONCLUSION 93

4.4 Conclusion

Several fatigue life prediction models exist. A classic Weibull approach isused in this work because of its high efficiency to simplicity ratio. Differentstress criteria are briefly presented. They lead to similar results, thus thesimplest one is preferred. An approximate calculation is proposed referred toas pseudo-transient. It is based on the conclusions from chapter 3 concerningthe pressure in elliptical contacts and during the contact crossing. Theaccuracy of this approximation is satisfactory and it allows a substantialtime and memory saving.

The maximum stress due to the indent is directly linked to the additionalpressure amplitude. Moreover, the risk integral is studied. A predictivemodel is proposed and a good correlation is obtained. The risk integral iswritten as a function of the stressed volume, the maximum stress value andthe infinite life limit. This model allows the prediction of the risk integralas a function of the indent geometry and the operating conditions.

Finally, a glimpse of the experimental work concerning indentation mea-surement and life prediction model adjustment is shown. Some results con-cerning natural indentation in a tapered roller bearing are presented, espe-cially statistical results on indent geometry. An adjustment procedure usedfor indented bearing life tests is briefly mentioned.

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94 CHAPTER 4. INDENTED CONTACT STRESS AND LIFE

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Chapter 5

General conclusion

In industrial applications, lubricant pollution can not be avoided. Controlledmanufacturing processes or lubricant filters can limit this pollution. Materialresistance to pollution is also constantly improved. However, the necessity topredict the life reduction due to lubricant pollution persists. The lubricantfilm thickness is usually much less than the particle size. When the particlesenter the contact, they are squashed between the two surfaces. The surfacesare deformed plastically creating an indent.

The indent geometry is complex to predict because it is a function ofmany parameters: particle nature and size, material properties, operatingconditions, particle mixing and lubricant supply. . . The aim of this work isto predict the indent severity as a function of the indent geometry and theoperating conditions. The surface indentation is supposed to be known froma surface measurement.

Chapter 2 presents the main equations and the simplifications used forthe contact pressure calculation (dry and EHL contacts) and for stress dis-tribution calculation. The indent analytical description is presented. Acompromise between description precision and calculation capability (time,stability, accuracy. . . ) is necessary. Multigrid techniques1 are used to solvecontact equations and to calculate deformation and risk integrals.

Chapter 3 studies the indented contact pressure. Dry and pure-rollinglubricated contacts are both studied. Three questions are asked: Whathappens when the indent traverses the contact? What is the influence ofthe contact ellipticity? What are the important parameters to model theindented contact pressure?

• For the range of parameters used in this work (indent geometry andoperating conditions), the centered indent calculation allows to pre-

1Appendix A presents an implementation of a transient EHL line relaxation solver.

95

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96 CHAPTER 5. GENERAL CONCLUSION

dict the entire indent crossing. The additional pressure distributionremains almost constant during the crossing2.

• Ellipticity hardly modifies the additional pressure for sufficiently smallindents (small force balance perturbation). Some elliptical results arepresented.

• An analytical model is presented for dry contacts and outlines theimportant phenomena and parameters3. The indent slope and theshoulder geometry are the important parameters. The lubricated con-tact study is based on the amplitude reduction theory which has beenextended to shallow indents. The lubrication substantially modifiesthe pressure perturbations, from a complete damping to almost nochanges. Some curves are presented to predict the EHL indented con-tact pressure perturbations.

Sliding, even limited, has an important detrimental effect. Some re-sults are briefly presented. No detailed study is proposed as indentsthat are created under rolling sliding conditions have complex geome-tries (not correctly described by the axisymmetrical model).

Chapter 4 presents some indented contact stress results and life predic-tion calculations. A brief study concerning the choice of the stress criterionand convenient application of some results of chapter 3 are presented. Asimple Von Mises stress criterion is used; however, more sophisticated mul-tiaxial criteria give similar results.

The maximum stress varies linearly with the additional pressure ampli-tude. A Weibull life prediction approach is presented; some curves for therisk integral prediction are presented as a function of the operating condi-tions, the indent geometry and the material infinite life limit.

Some results on natural indentation consecutive to a standard indenta-tion test with 90−125 µm ductile particles are shown. The sharpest indentsare found to be the smallest. The indentation geometry statistics for rollingelement bearings is specific because of the numerous overrollings. A simpli-fied example for a gear box tapered roller bearing is also presented.

A life prediction tool for (indented) rolling element bearings has beendeveloped and implemented. First, the surface topography analysis hasbeen improved and is still being studied; second, the specific kinematic andloading cycle has been accounted for.

2neglecting the outlet Newtonian pressure spike for EHL contacts3some details in appendix B

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97

First and foremost, the next step is an experimental validation and cor-relation. Such validations are time consuming and expensive. The testsshould at the same time show statistically significant differences and respectrealistic conditions. Typically, one should obtain simultaneously the samedamage type as observed in real applications, show statistically significant(quite large) differences between tested groups and be feasible on a practicaltime scale (especially for “clean” condition reference tests).

A very ambitious project is a more physical RCF model, where mate-rial RCF properties are extracted from the microstructural behavior. Itis certainly the4 Achilles heel of current RCF prediction in rolling elementbearings.

4The reader can substitute “the” by “the principal”; however, historically speakingAchilles had only one.

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98 CHAPTER 5. GENERAL CONCLUSION

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Appendix A

Transient EHL solver

implementation

Discrete equations

ξ =ρH3

ηλ(A.1)

ξi+1/2,j =ξi,j + ξi+1,j

2rh2

X

ξi−1/2,j =ξi−1,j + ξi,j

2rh2

X

ξi,j+1/2 =ξi,j + ξi,j+1

2rh2

Y

ξi,j−1/2 =ξi,j−1 + ξi,j

2rh2

Y

Σ ξ = ξi−1/2,j + ξi+1/2,j + ξi,j−1/2 + ξi,j+1/2 (A.2)

The NU2 scheme is composed of two schemes depending on the hX/hT

ratio. The operator Lh is defined in the implementation 1 as:

• if hX 6 hT one gets2:

Lh⟨

uh⟩

=rh2X

(

ξi+1/2,jPi+1,j −(

ξi+1/2,j + ξi−1/2,j

)

Pi,j − ξi−1/2,jPi−1,j

)

+rh2Y

(

ξi,j+1/2Pi,j+1 −(

ξi,j+1/2 + ξi,j−1/2

)

Pi,j − ξi,j−1/2Pi,j−1

)

− (rhX − rhT ) (1.5 ρHi,j − 2.0 ρHi−1,j + 0.5 ρHi−2,j)

−rhT 1.5 ρHi,j (A.3)

1to clarify the notation a little, the third subscript k related to the time step is omittedfor the terms corresponding to the current time step k

2for the finest level if hT = hX

111

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112 APPENDIX A. TRANSIENT EHL SOLVER IMPLEMENTATION

• if hX > hT one gets3:

Lh⟨

uh⟩

=rh2X

(

ξi+1/2,jPi+1,j −(

ξi+1/2,j + ξi−1/2,j

)

Pi,j − ξi−1/2,jPi−1,j

)

+rh2Y

(

ξi,j+1/2Pi,j+1 −(

ξi,j+1/2 + ξi,j−1/2

)

Pi,j − ξi,j−1/2Pi,j−1

)

−rhX 1.5 ρHi,j − (rhT − rhX) 1.5 ρHi,j (A.4)

All the terms which correspond to previous time step are put in theright hand side of the equation. The residual terms for FAS scheme are notchanged and coarsening routine is not changed. The right hand memberreads:

• if hX 6 hT one gets:

fhi,j = rhT (−2.0 ρHi−1,j,k−1 + 0.5 ρHi−2,j,k−2)

(A.5)

• if hX > hT one gets:

fhi,j = rhX (−2.0 ρHi−1,j,k−1 + 0.5 ρHi−2,j,k−2) +

(rhT − rhX) (−2.0 ρHi,j,k−1 + 0.5 ρHi,j,k−2) (A.6)

Gauss Seidel line relaxation

One can define the terms dH(k)X and dH

(k)T . They will be used to build the

line relaxation matrix.

3for all the coarser levels

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113

• if hX 6 hT one gets:

dH(−3)X = (rhX − rhT ) (1.5 ρi,jK3,0 − 2.0 ρi−1,jK2,0 + 0.5 ρi−2,jK1,0)

dH(−3)T = rhT 1.5 ρi,jK3,0

dH(−2)X = (rhX − rhT ) (1.5 ρi,jK2,0 − 2.0 ρi−1,jK1,0 + 0.5 ρi−2,jK0,0)

dH(−2)T = rhT 1.5 ρi,jK2,0

dH(−1)X = (rhX − rhT ) (1.5 ρi,jK1,0 − 2.0 ρi−1,jK0,0 + 0.5 ρi−2,jK1,0)

dH(−1)T = rhT 1.5 ρi,jK1,0

dH(0)X = (rhX − rhT ) (1.5 ρi,jK0,0 − 2.0 ρi−1,jK1,0 + 0.5 ρi−2,jK2,0)

dH(0)T = rhT 1.5 ρi,jK0,0

dH(+1)X = (rhX − rhT ) (1.5 ρi,jK1,0 − 2.0 ρi−1,jK2,0 + 0.5 ρi−2,jK3,0)

dH(+1)T = rhT 1.5 ρi,jK1,0

dH(+2)X = (rhX − rhT ) (1.5 ρi,jK2,0 − 2.0 ρi−1,jK3,0 + 0.5 ρi−2,jK4,0)

dH(+2)T = rhT 1.5 ρi,jK2,0 (A.7)

• if hX > hT one gets:

dH(−3)X = rhX 1.5 ρi,jK3,0

dH(−3)T = (rhT − rhX)1.5 ρi,jK3,0

dH(−2)X = rhX 1.5 ρi,jK2,0

dH(−2)T = (rhT − rhX)1.5 ρi,jK2,0

dH(−1)X = rhX 1.5 ρi,jK1,0

dH(−1)T = (rhT − rhX)1.5 ρi,jK1,0

dH(0)X = rhX 1.5 ρi,jK0,0

dH(0)T = (rhT − rhX)1.5 ρi,jK0,0

dH(+1)X = rhX 1.5 ρi,jK1,0

dH(+1)T = (rhT − rhX)1.5 ρi,jK1,0

dH(+2)X = rhX 1.5 ρi,jK2,0

dH(+2)T = (rhT − rhX)1.5 ρi,jK2,0 (A.8)

The matrix Aj in equation 2.39 reads:

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114 APPENDIX A. TRANSIENT EHL SOLVER IMPLEMENTATION

Aji,i−3 = −dH

(−3)X − dH

(−3)T

Aji,i−2 = −dH

(−2)X − dH

(−2)T

Aji,i−1 = −dH

(−1)X − dH

(−1)T + ξi−1/2,j

Aji,i = −dH

(0)X − dH

(0)T − Σ ξ

Aji,i+1 = −dH

(+1)X − dH

(+1)T + ξi+1/2,j

Aji,i+2 = −dH

(+2)X − dH

(+2)T (A.9)

Jacobi distributive line relaxation

dK0 = K0,0 −1

4(2K1,0 + 2K0,1)

dK1 = K1,0 −1

4(K2,0 + K0,0 + 2K1,1)

dK2 = K2,0 −1

4(K3,0 + K1,0 + 2K2,1)

dK3 = K3,0 −1

4(K4,0 + K2,0 + 2K3,1)

dK4 = K4,0 −1

4(K5,0 + K3,0 + 2K4,1) (A.10)

• if hX 6 hT one gets:

dH(−3)X = (rhX − rhT ) (1.5 ρi,jdK3 − 2.0 ρi−1,jdK2 + 0.5 ρi−2,jdK1)

dH(−3)T = rhT 1.5 ρi,jdK3

dH(−2)X = (rhX − rhT ) (1.5 ρi,jdK2 − 2.0 ρi−1,jdK1 + 0.5 ρi−2,jdK0)

dH(−2)T = rhT 1.5 ρi,jdK2

dH(−1)X = (rhX − rhT ) (1.5 ρi,jdK1 − 2.0 ρi−1,jdK0 + 0.5 ρi−2,jdK1)

dH(−1)T = rhT 1.5 ρi,jK1,0

dH(0)X = (rhX − rhT ) (1.5 ρi,jdK0 − 2.0 ρi−1,jdK1 + 0.5 ρi−2,jdK2)

dH(0)T = rhT 1.5 ρi,jdK0

dH(+1)X = (rhX − rhT ) (1.5 ρi,jdK1 − 2.0 ρi−1,jdK2 + 0.5 ρi−2,jdK3)

dH(+1)T = rhT 1.5 ρi,jdK1

dH(+2)X = (rhX − rhT ) (1.5 ρi,jdK2 − 2.0 ρi−1,jdK3 + 0.5 ρi−2,jdK4)

dH(+2)T = rhT 1.5 ρi,jdK2 (A.11)

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115

• if hX > hT one gets:

dH(−3)X = rhX 1.5 ρi,jdK3

dH(−3)T = (rhT − rhX)1.5 ρi,jdK3

dH(−2)X = rhX 1.5 ρi,jdK2

dH(−2)T = (rhT − rhX)1.5 ρi,jdK2

dH(−1)X = rhX 1.5 ρi,jdK1

dH(−1)T = (rhT − rhX)1.5 ρi,jdK1

dH(0)X = rhX 1.5 ρi,jdK0

dH(0)T = (rhT − rhX)1.5 ρi,jdK0

dH(+1)X = rhX 1.5 ρi,jdK1

dH(+1)T = (rhT − rhX)1.5 ρi,jdK1

dH(+2)X = rhX 1.5 ρi,jdK2

dH(+2)T = (rhT − rhX)1.5 ρi,jdK2 (A.12)

The matrix Aj in equation 2.42 reads:

Aji,i−3 = −dH

(−3)X − dH

(−3)T

Aji,i−2 = −dH

(−2)X − dH

(−2)T − 1

4ξi−1/2,j

Aji,i−1 = −dH

(−1)X − dH

(−1)T + ξi−1/2,j +

1

4Σ ξ

Aji,i = −dH

(0)X − dH

(0)T − 5

4Σ ξ

Aji,i+1 = −dH

(+1)X − dH

(+1)T + ξi+1/2,j +

1

4Σ ξ

Aji,i+2 = −dH

(+2)X − dH

(+2)T − 1

4ξi+1/2,j (A.13)

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116 APPENDIX A. TRANSIENT EHL SOLVER IMPLEMENTATION

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Appendix B

Dry contact analytical model

The approximation of the hole and shoulder geometry influences additionalpressure results. No study concerning the best approximation has beenmade. The aim was not to get the most accurate prediction of the additionalpressure, but to introduce a physical approach of the shoulder influence inzone A, and to account for the zero pressure in zone B. The functions ofthe geometrical approximation functions are briefly reported below. Thesefunctions have no simple analytical expressions, but are easy to evaluatenumerically.

The indent geometry and its derivatives read:

R = −de−K

r2

φ2cos

(

πr

φ

)

(B.1)

R′ =∂R∂r

= de−K

r2

φ2

2Kr

φ2cos

(

πr

φ

)

φsin

(

πr

φ

)

(B.2)

R′′=∂2R∂r2

=2dK

φ2e−K

r2

φ2

cos

(

πr

φ

) (

1 − 2Kr2

φ2

)

− 2πr

φsin

(

πr

φ

)

(B.3)

117

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118 APPENDIX B. DRY CONTACT ANALYTICAL MODEL

Shoulder approximation

To calculate the shoulder additional pressure amplitude , a parabolic ap-proximation of the shoulders has been used see figure B.3 and equation B.4.The shoulder approximation reads :

Figure B.1: Shoulder model diagram

Rapprox = − 1

ρh

r

2

2+ sh (B.4)

The values rh, sh and ρh reads :

R′ (rh) = 0

rh ∈ ]φ

2

2[

tan (rh) = −2K

π2rh

rh =π rh

φ

rh ∈ ]1

2

3

2[

The equation above is evaluated numerically, and the solution enablesto deduce the numerical values below :

rh = β(K)φ (B.5)

sh = R (rh) = γ(K) d (B.6)

ρh =1

R′′ (rh)= µ(K)

φ

d

2

(B.7)

ν(K) =

γ(K)

µ(K)(B.8)

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119

Two parameters which are function of K are used to calculate ν(K).The radius of curvature of the shoulder ρh appears through the parameterµ(K) and γ(K) corresponds to the shoulder height sh. The factor ν(K) isgiven in figure B.2. It is used to calculate the shoulder influence.

3 4 5 6 7 80.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

K

ν

Figure B.2: ν as a function of K

Hole approximation

To calculate the hole additional pressure amplitude , the choice has beenmade to represent the hole with a parabolic approximation see figure B.3.The hole approximation reads :

Rapprox = − 1

ρb

r

2

2+ d (B.9)

Two parameters which are function of K are used to compute ζ(K). Theradius of curvature at the hole bottom ρb appears through the parameterλ(K), and β(K) is related to the radius rh.

ρb = λ(K)φ

d

2

(B.10)

ζ(K) =2√

λ(K)

β(K)(B.11)

if R(φ/4)= Rapprox(φ/4)

λ(K) = 1

32

(

1 − e−K/16

√2

)

When an approximation like the one represented figure B.3 is used, ζ(K)is almost constant and equal to 0.78.

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120 APPENDIX B. DRY CONTACT ANALYTICAL MODEL

−1 −0.5 0 0.5 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2K=3model

−1 −0.5 0 0.5 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2K=8model

Figure B.3: Hole approximation with R(Φ/4) = Rapprox(Φ/4)

Zone A coefficient adjustment

The coefficients of proportionality m and n of the shoulder and the holeinfluence in equation 3.5 are adjusted in zone A using the method of leastsquares see equations B.12, B.13.

x = ∆P ′hole

y = ∆P ′shoulders

f = ∆Pnumtot

T =∑

x2i ·

y2i −

(

xi yi

)2

n =(

y2i ·

fi xi −∑

xi yi ·∑

fi yi

)

/ T (B.12)

m =(

x2i ·

fi yi −∑

xi yi ·∑

fi xi

)

/ T (B.13)

(B.14)

Zone B coefficient adjustment

As mentioned before, the ratio between deformed and initial indent depth Fis directly linked to the indent slope D/Φ. The zero pressure in the indenthole appears for a slope of 0.25. Moreover, an horizontal asymptote appearswhen the slope is large because the hole will be less and less deformed.When the slope decreases to 0.25, a second asymptote correctly matches

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121

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

D/Φ

Dd/D

i

numericalmodel

ZO

NE

A

ZO

NE

B

Figure B.4: Ratio between deformed and initial indent depth Dd/Di as afunction of the indent slope D/Φ

the numerical results. The parameters k and θ of equation B.15 have beenadjusted accordingly. The final equations reads:

F =Dd

Di

θ = 1.1

k = 2.1

F =k (D/Φ − 0.25)

θ

1 + k (D/Φ − 0.25)θfor D/Φ > 0.25 (B.15)

F = 0 for D/Φ 6 0.25

∆Ptot = nD

Φζ(K) (1 − sFω) + m

D

Φν(K) (B.16)

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122 APPENDIX B. DRY CONTACT ANALYTICAL MODEL

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Appendix C

Line contact stress

For infinitely elongated Hertzian contact without friction, the stress distri-bution in the material has a simple analytical expression C.1, see Smith andLiu [104]. The authors detail also a more complicated set of equations toaccount for Hertzian contact with friction. Elliptical contacts were studiedby Sackfields and Hills [100, 101].

C1 = (1 + X)2 + Z2

C2 = (1 − X)2 + Z2

ψ =π

C1

1 −√

C2

C1√

C2

C1

2

C2

C1+

C1 + C2 − 4

C1

ψ =π

C1

1 +

C2

C1√

C2

C1

2

C2

C1+

C1 + C2 − 4

C1

σxx = −Z

π

(

ψ (1 + 2X2 + 2Z2) − 2 π − 3 Xψ)

σyy = ν (σxx + σzz)

σzz = −Z

π

(

ψ − Xψ)

σxz = −Z2ψ

π(C.1)

123

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124 APPENDIX C. LINE CONTACT STRESS

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List of Figures

1.1 Tapered roller bearing . . . . . . . . . . . . . . . . . . . . . . 6

1.2 Struggle against friction . . . . . . . . . . . . . . . . . . . . . 7

1.3 Typical EHL pressure distribution and film thickness alongthe rolling direction . . . . . . . . . . . . . . . . . . . . . . . 8

1.4 M50 particles [119] (distribution 0 − 100 µm) . . . . . . . . . 9

1.5 SiC particles [119] (mean size 45 µm) . . . . . . . . . . . . . 9

1.6 Spalling due to an artificial indent [82] . . . . . . . . . . . . . 10

1.7 Pollution modeling in a real device . . . . . . . . . . . . . . . 12

1.8 Microstructure of a case carburized quenched 16NiCrMo13steel [108] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.1 Contacting body geometries – real geometry (left) and equiv-alent geometry (right) [28] . . . . . . . . . . . . . . . . . . . 18

2.2 Comparison of experimental (left) and numerical (right) re-sults of a transverse ridge under pure-rolling, experimentalresult by Kaneta [71] and numerical result by Lubrecht [110] 22

2.3 Stress meshing . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.4 Ductile particle indent 3D mapping by Cogdell [23] . . . . . 25

2.5 Ductile particle indent [119] . . . . . . . . . . . . . . . . . . 25

2.6 Indent profiles for different decay coefficients K from 3 to 8,D = 1, Φ = 1 . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.7 Example of a V (ν1, ν2)-cycle with 4 levels [115] . . . . . . . . 32

2.8 FMG with one V-cycle [115] . . . . . . . . . . . . . . . . . . . 33

2.9 Additional pressure amplitude definition . . . . . . . . . . . 36

2.10 Error in ∆PDRY for a dry contact as a function of nX . Indentgeometry Φ = 0.25 0.5, D/Φ = 2, K = 4 . . . . . . . . . . . 37

2.11 Error in ∆PEHLLAT for a stationary EHL contact M = 200, L =

19 as a function of nX . Groove geometry Φ = 0.125 0.25 0.5,D/Φ = 1/4, K = 4 . . . . . . . . . . . . . . . . . . . . . . . . 38

2.12 Error in ∆PEHLLAT for a stationary EHL contact M = 800, L =

8 as a function of nX . Groove geometry Φ = 0.125 0.25 0.5,D/Φ = 1/4, K = 4 . . . . . . . . . . . . . . . . . . . . . . . . 38

125

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126 LIST OF FIGURES

2.13 Error in ∆PEHLREAR ∆PEHL

LAT ∆PEHLCENT for a transient EHL con-

tact M = 200, L = 19 as a function of nX . Indent geometryΦ = 0.25 0.5, D/Φ = 1, K = 4 . . . . . . . . . . . . . . . . . 39

2.14 Error in ∆PEHLREAR ∆PEHL

LAT ∆PEHLCENT for a transient EHL con-

tact M = 800, L = 8 as a function of nX . Indent geometryΦ = 0.25 0.5, D/Φ = 1, K = 4 . . . . . . . . . . . . . . . . . 40

2.15 Example of a stress distribution in a vertical plane for a drycontact with a centered indent . . . . . . . . . . . . . . . . . 41

2.16 Error in σV Mmax for a dry contact as a function of nX . Indent

geometry Φ = 0.25 0.5, D/Φ = 2, K = 4 . . . . . . . . . . . 42

2.17 Error in ID for a dry contact as a function of nX . Indentgeometry Φ = 0.25 0.5, D/Φ = 2, K = 4 . . . . . . . . . . . 42

3.1 2D pressure distribution in an indented dry contact Φ = 0.5,K = 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.2 Additional pressure amplitude definition for dry contacts Φ =0.5, K = 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.3 Indent traversing a dry contact Φ = 0.5 . . . . . . . . . . . . 47

3.4 Dry contact pressure distribution around an indent as a func-tion of the ellipticity Φ = 0.5 . . . . . . . . . . . . . . . . . . 50

3.5 Zone A - B typical pressure distribution . . . . . . . . . . . . 51

3.6 Analytical model pressure distribution . . . . . . . . . . . . . 52

3.7 Shoulder approximation . . . . . . . . . . . . . . . . . . . . . 53

3.8 Ratio between the additional pressure amplitude and the in-dent slope ∆P

D/Φ in zone A (completely flattened indents) as a

function of the indent slope D/Φ and the decay coefficient K 54

3.9 Example of deformed indent geometry in zone B (partiallyflattened indent) . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.10 Ratio between deformed and initial indent depth Dd/Di as afunction of the indent slope D/Φ . . . . . . . . . . . . . . . . 56

3.11 Ratio between the additional pressure amplitude and the in-dent slope ∆P

D/Φ in zone B (partially flattened indent) as a

function of the indent slope D/Φ and the decay coefficient K 57

3.12 Full numerical additional pressure versus analytical predic-tion for Φ = 0.25 − 0.5, D/Φ = 0 − 2 and K = 3 − 8 . . . . . 57

3.13 Additional pressure amplitude definition for EHL contactsΦ = 0.5, M = 400, L = 5 . . . . . . . . . . . . . . . . . . . . 58

3.14 Indent traversing an EHL contact M = 400, L = 5 . . . . . . 59

3.15 Pressure distribution around an indent when varying the el-lipticity in an EHL contact, Φ = 0.5, M = 400, L = 10 . . . 60

3.16 Additional pressure amplitude over the rear shoulder in ehlcontacts compared to a dry contact when varying the meanspeed, Φ = 0.5 . . . . . . . . . . . . . . . . . . . . . . . . . . 63

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LIST OF FIGURES 127

3.17 Relative initial slope S∆PEHLREAR/S∆PDRY as a function of

the dimensionless indent diameter parameter ∇Φ, for M =100 − 1200, L = 3 − 19, Φ = 0.25 − 1.0, K = 3 . . . . . 65

3.18 Relative initial slope S∆PEHLFRONT /S∆PDRY as a function of

the dimensionless indent diameter parameter ∇Φ, for M =100 − 1200, L = 3 − 19, Φ = 0.25 − 1.0, K = 3 . . . . . 65

3.19 Relative initial slope S∆PEHLCENT /S∆PDRY as a function of

the dimensionless indent diameter parameter ∇Φ, for M =100 − 1200, L = 3 − 19, Φ = 0.25 − 1.0, K = 3 . . . . . 66

3.20 Relative initial slope S∆PEHL/S∆PDRY as a function ofthe dimensionless indent diameter parameter ∇Φ, for M =100 − 1200, L = 3 − 19, Φ = 0.25 − 1.0, K = 3 . . . . . 67

3.21 Relative slope S∆PEHLREAR/S∆PDRY as a function of the di-

mensionless indent diameter parameter ∇Φ and of the shoul-der parameter K, for M = 100 − 1200, L = 3 − 19,Φ = 0.25 − 1.0 . . . . . . . . . . . . . . . . . . . . . . . . . 68

3.22 Additional pressure amplitude over the front shoulder ∆PEHLFRONT /PREF

for sharper indents K = 4. Blue curves for Φ = 0.25, redcurves for Φ = 0.5 and green curves for Φ = 1 . . . . . . . . 69

3.23 Additional pressure amplitude over the rear shoulder ∆PEHLREAR/PREF

for sharper indents K = 4. Blue curves for Φ = 0.25, redcurves for Φ = 0.5 and green curves for Φ = 1 . . . . . . . . 69

3.24 Additional pressure amplitude in the indent hole ∆PEHLCENT /PREF

for sharper indents K = 4. Blue curves for Φ = 0.25, redcurves for Φ = 0.5 and green curves for Φ = 1 . . . . . . . . 70

3.25 ∆PEHLFRONT /S∆PEHL

FRONT FIT and ∆PEHLREAR/S∆PEHL

REAR FIT fordeeper indents. M = 100 − 1200, L = 3 − 19, Φ = 0.25 − 1and K = 3 − 8 . . . . . . . . . . . . . . . . . . . . . . . . . . 71

3.26 Full numerical prediction versus model predictions of the ad-ditional pressure amplitude over the front shoulder for M =100 − 1200, L = 3 − 19, Φ = 0.25 − 1, D/Φ = 0 − 2 andK = 3 − 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

3.27 Full numerical prediction versus model prediction of the ad-ditional pressure amplitude over the rear shoulder for M =100−1200, L = 3−19, Φ = 0.25−1, D/Φ = 0−2 and K = 3−8 72

3.28 Pressure distribution for a centered indent for different slideto roll ratios 0%, +2%, +4%. Indent on the fast surface Φ =0.4, M = 500, L = 8 . . . . . . . . . . . . . . . . . . . . . . . 73

3.29 Indent traversing a +4% rolling sliding EHL contact. Indenton the fast surface Φ = 0.4, M = 500, L = 8 . . . . . . . . . 74

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128 LIST OF FIGURES

4.1 Stress distribution σV M/σV MS max at different indent locations

a-d, and stress amplitude history e-f in the vertical planealong the rolling direction for a circular dry contact (e smooth,f dented), Φ = 0.5, K = 3 . . . . . . . . . . . . . . . . . . . . 79

4.2 Indented contact stress amplitude history σPA/σPAS max for Pa-

padopoulos’ multiaxial criterion for a circular dry contact,Φ = 0.5, K = 3 . . . . . . . . . . . . . . . . . . . . . . . . . . 81

4.3 Indented contact stress amplitude history σV MPT /σV M

S max in thevertical plane along the axis X for a pseudo-transient calcu-lation for a circular dry contact, Φ = 0.5, K = 3 . . . . . . . 81

4.4 Maximum stress TD max/TS max as a function of the additionalpressure amplitude ∆P/PREF for dry contacts, Φ = 0.25 −1.0, D/Φ = 0.25 − 2 and K = 3 − 8 . . . . . . . . . . . . . . 84

4.5 Dimensionless risk integral ID/IS as a function of the indentgeometry Φ, D/Φ and K and of the dimensionless infinite lifelimit T∞/TS max for dry contacts . . . . . . . . . . . . . . . . 85

4.6 Adjustment functions qK(K) (plot a) and qT∞(T∞/TS max)

(plot b), dimensionless risk integral IS/IS T∞=0 as a functionof the dimensionless infinite life limit T∞/TS max for dry con-tacts (plot c) . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

4.7 Full numerical dimensionless risk integral as a function of themodel prediction for dry contacts, c = 6.3−11.3, T∞/TS max =0.3 − 0.7, Φ = 0.25 − 1.0, K = 3 − 8 . . . . . . . . . . . . . . 87

4.8 Indent slope d/φ (left) and indent diameter φ (right) cumu-lative distribution from a standard pollution test . . . . . . . 88

4.9 Indent diameter as a function of the indent slope (left) andprobability of every indent slope class (right) from a topog-raphy measurement of a standard pollution test . . . . . . . 89

4.10 Dimensionless risk integral ID/IS as a function of the indentgeometry for a dry contact applied to a topography measure-ment of a standard pollution indentation test (90 − 125 µmductile particles) . . . . . . . . . . . . . . . . . . . . . . . . . 91

4.11 Dimensionless risk integral ID/IS as a function of the in-dent geometry for an EHL contact applied to a topogra-phy measurement of a standard pollution indentation test(90 − 125 µm ductile particles) . . . . . . . . . . . . . . . . . 91

B.1 Shoulder model diagram . . . . . . . . . . . . . . . . . . . . 118B.2 ν as a function of K . . . . . . . . . . . . . . . . . . . . . . . 119B.3 Hole approximation with R(Φ/4) = Rapprox(Φ/4) . . . . . . 120B.4 Ratio between deformed and initial indent depth Dd/Di as a

function of the indent slope D/Φ . . . . . . . . . . . . . . . . 121

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List of Tables

1.1 Particle origins and consequences on indentation [119] . . . . 9

2.1 Experimental observation of the indent diameter, depth andslope as function of the particle diameter [119] . . . . . . . . 26

3.1 Difference in the additional pressure amplitude during indentcrossing for dry contacts as a function of indent diameter Φand indent slope D/Φ for K = 4 . . . . . . . . . . . . . . . . 48

3.2 Difference in the additional pressure amplitude for two ellip-ticities κ = 1.00 and κ = for dry contacts as a function of theindent diameter Φ and the indent slope D/Φ for K = . . . . 51

3.3 Evolution of the dry contact additional pressure amplitude∆PDRY /PREF as a function of the indent slope D/Φ, theshoulder geometry K and the indent diameter Φ . . . . . . . 53

3.4 Difference in the additional pressure amplitude over the rearshoulder during indent crossing for EHL contacts as a func-tion of indent diameter Φ and indent slope D/Φ for K = 4,M = 400, L = 5 . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.5 Difference in the additional pressure amplitude over the rearshoulder for two ellipticities κ = 1.0 and κ = 0.2 for EHLcontacts =

∣∆PEHLκ=1.0 − ∆PEHL

κ=0.2

∣ /∆PEHLκ=1.0 as a function of the

operating conditions M , L, the indent diameter Φ and theindent slope D/Φ for K = 4 . . . . . . . . . . . . . . . . . . 61

4.1 Difference between different stress criteria as a function ofthe indent diameter Φ, the indent slope D/Φ and the decaycoefficient K for dry contacts . . . . . . . . . . . . . . . . . . 83

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